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+page_content='PR]  12 Jan 2023  Singular SPDEs on Homogeneous Lie Groups  Avi Mayorcas1 and Harprit Singh2  1Institute of Mathematics, Technische Universität Berlin, Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' des 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Juni 136, 10587 Berlin, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Email: avimayorcas@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
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+page_content=' ORCID iD: 0000-0003-4133-9740  2Department of Mathematics, Imperial College London, South Kensington Campus, SW7 2AZ, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
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+page_content=' ORCID iD: 0000-0002-9991-8393  January 13, 2023  Abstract  The aim of this article is to extend the scope of the theory of regularity structures in order  to deal with a large class of singular SPDEs of the form  ∂tu = Lu +F(u,ξ) ,  where the differential operator L fails to be elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' This is achieved by interpreting the  base space Rd as a non-trivial homogeneous Lie group G such that the differential oper-  ator ∂t −L becomes a translation invariant hypoelliptic operator on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Prime examples  are the kinetic Fokker-Planck operator ∂t − ∆v − v · ∇x and heat-type operators associ-  ated to sub-Laplacians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' As an application of the developed framework, we solve a class  of parabolic Anderson type equations  ∂tu =  �  i  X 2  i u +u(ξ−c)  on the compact quotient of an arbitrary Carnot group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Keywords: Regularity structures, homogeneous Lie groups, hypoelliptic operators, stochastic  partial differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  2020 MSC: 60L30 (Primary);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 60H17, 35H10, 35K70 (Secondary)  CONTENTS  1 Introduction  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1  Related Literature .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
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+page_content='  55  5 A Regularity Structure for Anderson Equations  57  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1  Smooth Models .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
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+page_content='  59  6 The Anderson Equation on Stratified Lie Groups  59  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  63  1 INTRODUCTION  The theory of regularity structures, [Hai14], provides a framework for the study of subcritical  stochastic partial differential equations (SPDEs) of the form  ∂tu −Lu = F(u,ξ) ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1)  when the operator L is uniformly elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' This article extends the theory of regularity struc-  tures in order to solve equations where the differential operator stems from a large class of hy-  poelliptic operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' This is achieved by building on the fundamental idea of Folland [Fol75]  to reinterpret the differential operator in question as a differential operator on a homoge-  neous Lie group and extending the theory of regularity structures to these non-commutative  spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  The treatment of singular SPDE via the theory of regularity structures can be divided into  two parts, an analytic step and an algebraic and stochastic step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The first, analytic step, is to introduce the notions of a regularity structure, models and  modelled distributions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' generalised Taylor expansions of both functions and distribu-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Given this set-up, sufficient analytic tools are then developed to allow for the  study of abstract, fixed-point equations in the spaces of modelled distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Cru-  cially, a reconstruction operator maps modelled distributions to genuine distributions  (or functions) on the underlying domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For the case of constant coefficient, hypoel-  liptic equations on compact quotients of Euclidean domains, this aspect of the theory  was already worked out in full generality in [Hai14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The second step, which was carried out in a case by case basis in [Hai14], was then  fully automated in the subsequent works [CH16, BHZ19, BCCH21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In [BHZ19] the au-  thors construct concrete (equation-dependent) regularity structures and models to-  gether with a large enough renormalisation group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Then, in [BCCH21], the question  of how this renormalisation group acts on the SPDE in question was answered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Con-  vergence of renormalised models for an extremely large class of noises, known as the  BPHZ theorem, was proved in [CH16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  In much of the above work translation invariance on Rd, of both the equation and the driving  noise, plays a major role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In this paper we will instead work with equations that are translation  invariant with respect to a homogeneous Lie group G (Euclidean spaces being special case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  The bulk of this this paper is dedicated to implementing the first analytic part of the theory  in the case when the underlying space is a general (non-Abelian) homogeneous Lie group,  the second step is then carried out for a specific class of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' While many of the key  ideas from [Hai14] carry over to our setting, we encounter a number of significant devia-  tions from the arguments presented therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The main cause of these deviations are the non-  commutative structure of the base space and the fact that the notion of Taylor expansion, a  crucial element of the theory, heavily depends on the underlying group structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' While this  article aims at being relatively self-contained, we focus mainly on these required deviations  from [Hai14] and do not reproduce arguments which carry over directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  The central motivation for this extension to homogeneous Lie groups is to allow for the  study of singular SPDE with linear part given by a hypoelliptic operator, which may fail to  be uniformly parabolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Two motivating examples are the heat operator associated to the  sub-Laplacian on the Heisenberg group and the kinetic Fokker-Planck operator on R×R2n,  u(t,x, y,z) �→ ∂tu −(∆x +∆y +(x2 + y2)∆z)u  and  u(t,x,v) �→ ∂tu −∆vu − v ·∇xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2)  We will carry these two operators and their associated homogeneous Lie groups, throughout  the paper as working examples of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In the final sections we develop a full solution  theory for Anderson type equations associated to sub-Laplacians on general stratified Lie  groups (Carnot groups), of which the Heisenberg group is a well-studied example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  More generally, however, the analytic results of this paper apply to any differential operator,  ¯L on Rd−1, such that the following combination of results by Folland applies to L := ∂t − ¯L,  with respect to some homogeneous Lie group structure on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 ([Fol75, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 & Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Let G be a homogeneous Lie group of homoge-  neous dimension |s| and L be a left-translation invariant (with respect to G), homogeneous  3  differential operator of degree β ∈ (0,|s|) on G, such that L and its adjoint L∗ are both hypoel-  liptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Then there exists a unique homogeneous distribution K , of order β−|s| such that for any  distribution, ϕ ∈ D′(G)  L(ϕ∗K ) = (Lϕ)∗K = ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  where the convolution is with respect to the given Lie structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Referring to Section 2 for a more thorough discussion of homogeneous Lie groups and their  properties, we recall here that a differential operator D on Rd is called hypoelliptic, if for every  open subset Ω ⊂ Rd one has that,  Du ∈C ∞(Ω) ⇒ u ∈C ∞(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  The following celebrated result of Hörmander (almost) entirely classifies the family of second  order hypoelliptic operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2 ([Hö67, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Let r ≤ d and {Xi }r  i=0 be a collection of first order differential  operators (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' vector fields) on Rd and recursively define V1 = span{Xi : i = 0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=',r}, Vn+1 =  Vn ∪{[V,W ] : V ∈ V1, W ∈ Vn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' If a differential operator can be written in the form,  D =  r�  i=1  X 2  i + X0 +c,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3)  for some c ∈ R, and there exists an N ≥ 1 such that dim(VN) = d at every point in Ω ⊆ Rd then  D is hypoelliptic on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' By Froebenious’s theorem, [Fro77], if the condition of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2 fails in some  open set then D is not hypoelliptic on that set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' However, the statement is not truly sharp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  for example the Grushin operator ∂2  vv − v∂x is hypoelliptic, while the conditions of Theo-  rem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2 are violated on sets intersecting {v = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' On the other hand this type of exception  cannot occur if the sections Vi are continuous, since in this case it can be shown that the  map x �→ 1{dimVi(x)=d} is upper semi-continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  While Hörmander’s theorem gives an almost sharp characterisation of second order, hy-  poelliptic, operators it does not say much about fine properties of the fundamental solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  For example, it gives no direct route to a refined regularity theory for hypoelliptic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  This observation highlights the contribution of Folland’s theorem (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' since trans-  lation invariance allows access to many additional tools in the Euclidean setting, such as  harmonic analysis and singular integral methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Viewing the class of translation invariant  operators satisfying Folland’s theorem as analogous to constant coefficient, elliptic operators  on Rd a programme was successfully carried out in the works [FS74a, FS74b, Fol75, RS76], es-  tablishing a full Lp regularity theory for general, smooth coefficient, hypoelliptic operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  We refer to [Bra14, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 3] for a concise introduction to this programme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 RELATED LITERATURE  Non-translation invariant and non-uniformly elliptic SPDEs have been well-studied in the  classical, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' non-singular, regime and we do not attempt to present this large literature here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  4  We refer to standard texts such as [DPZ14, LR17, DKM+09, PR07] for a general overview and  references to more specific works contained therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' However, in the more specialised set-  ting of semi-linear SPDEs on homogeneous Lie groups, we mention the works [TV99, TV02,  PT10] which treat both hypoelliptic and parabolic SPDEs on some classes of Lie groups and  sub-Riemannian manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' A solution theory for conservative SPDEs based on the kinetic  Fokker–Planck equation (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4) in the Itô case was developed in [FFPV17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Using  the theory of paracontrolled calculus this was extended to the singular regime in [HZZZ21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  The kinetic Fokker–Plank operator and its associated homogeneous Lie group fall into the  analytic framework developed in this paper, however, we postpone a concrete application of  this theory towards a kinetic Anderson type equation to a future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Recently, in [BOTW22],  an Anderson type equation on the Heisenberg group with white in time and coloured in space  noise was studied using Itô calculus techniques, up the to the full sub-critical regime of the  noise regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We treat a closely related problem in the final sections of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' A fuller  discussion of the similarities and differences between the results of [BOTW22] and those of  our approach is postponed to Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Singular SPDEs with non-translation invariant, but uniformly parabolic or elliptic linear  part, have also been considered, especially using the recently developed pathwise techniques  of [Hai14, GIP15, OSSW21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Some of these works are discussed in more detail in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Quasilinear SPDEs have been studied using regularity structures, rough path based  methods and paracontrolled distribution theory, see [GH19b, OW19, BM22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Recently, an ap-  proach inspired by the theory of regularity structures, but technically distinct, has been devel-  oped in [OSSW21, LOT21, LO22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' SPDEs on domains with boundaries have been treated us-  ing both regularity structures and paracontrolled distribution based methods, [Lab19, GH19a,  CvZ21, GH21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Finally, a number of works have considered parabolic, singular SPDEs on Rie-  mannian manifolds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' a paracontrolled approach using the spectral decomposition associated  to the Laplace–Beltrami operator has been developed in [BB16, BB19, Ant22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' An approach  via regularity structures has been applied to the 2d parabolic Anderson equation on a Rie-  mannian manifold in [DDD19], while [BB21] develops some aspects of the general algebraic  structure required to treat non-translation invariant, uniformly parabolic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Finally,  the upcoming work [HSa] gives a comprehensive extension of regularity structures to sin-  gular SPDEs on Riemannian manifolds, with only the renormalisation of suitable stochastic  objects left to be done by hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2 A MOTIVATING CLASS OF EXAMPLES  In this paper we restrict ourselves to linear operators satisfying the criteria of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1  and use the Anderson equation as a main motivating example, although we stress that our  main analytic results apply in the full generality treated by [Hai14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The parabolic Anderson  model  ∂tu = ∆u −uξ,  u|t=0 = u0,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4)  on R+×Rd describes the conditional, expected density of particles, where each particle moves  according to an independent Brownian motion and branches at a rate proportional to the  random environment ξ, see [Kön16, CM94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Rigorously, this description is derived by discrete  approximations and it is well known that in the case ξ is a spatial white noise, when d = 2 and  5  d = 3 one needs to recentre the potential in order to obtain a non-trivial limit as the discreti-  sation is removed, [Hai14, HL15, HL18, GIP15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We point out that if the environment is also  allowed to depend on time and is for example, white in time, then martingale methods can  be used instead to develop a probabilistic solution theory, see [Wal86, Dal99, Dal01, Che15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  If one replaces the Brownian motions with a general diffusion,  dxt =  �  2  r�  i=1  Xi(xt)dW i  t ,  where the vector fields satisfy Hörmander’s rank condition (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2), then, formally the  conditional expected density of particles is described by the hypoelliptic Anderson equation,  ∂tu − ¯Lu := ∂tu −  r�  i=1  X 2  i u = uξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5)  When realisations of the environment are sufficiently singular then a pathwise solution the-  ory for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5) is expected to require renormalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Since the vector fields {Xi }r  i=1 satisfy Hör-  mander’s condition, the operator ¯L is smoothing and therefore in principle, an extension of  the theory of regularity structures should be applicable to find a renormalised, pathwise, so-  lution theory for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In Section 6 we apply the analytic tools developed in the paper to  demonstrate such an extension to the case where ¯L is the sub-Laplacian on a compact quo-  tient of stratified Lie group (Carnot group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  We show a result analogous to those of [Hai14, HL18], finding a notion of renormalised so-  lution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5), when ξ is a coloured periodic noise on a stratified Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' More precisely,  we show that when ξ is replaced with a mollified, recentred noise ξε − cε, for (specific) di-  verging constants {cε}ε∈(0,1), solutions uε to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5) converge in probability to a unique limit  independent of the specific choice of mollification-scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We stress that this result does  not cover the full subcritical regime of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The treatment of this full regime would require  an analogue of the BPHZ theorem on homogeneous Lie groups, see [CH16, HSb].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  A notable example of a stratified Lie group is the Heisenberg group, Hn ∼= R2n ×R, see Sec-  tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 for a description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In this case the collection of left-translation invariant vector fields,  which generate the associated Lie algebra are,  Ai(x, y,z) = ∂xi + yi∂z,  Bi(x, y,z) = ∂yi − xi∂z,  for i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  It is readily checked that the collections {Ai ,Bi}n  i=1 are left-translation invariant with respect  to the group action described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 and that one has C(x, y,z) := [Ai,Bi] = −2∂z for  all i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The associated sub-Laplacian is the linear differential operator,  ¯Lu =  n�  i=1  (A2  i +B2  i )u,  naturally extended to a heat type operator as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5), c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 for further  discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' A phenomenon of interest for Anderson equations is that of localisation, the con-  centration of the solution at large times, taking arbitrarily large values on islands of arbitrarily  small size, [CM94, Kön16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Since one expects the geometry of the underlying domain to have  6  effect on the emergence of this phenomenon, as also noted in [BOTW22] and since pathwise  approaches to the parabolic Anderson model have proved fruitful in studying finer proper-  ties of solutions, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' [AC15, HL15, HL18, GUZ19, Lab19, BDM22], it is our hope that the  tools developed herein may prove useful in analysing similar equations on more complex  domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The solution theory exposited in Section 6 applies to precisely this example on a  compact quotient of the Heisenberg group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3 OPEN PROBLEMS AND WIDER CONTEXT  As discussed above, translation invariant (and for example) parabolic operators on Euclidean  domains serve as a starting point from which non-translation invariant parabolic problems  on Euclidean domains as well as parabolic problems on Riemannian manifolds can be stud-  ied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' A somewhat parallel progression holds starting from translation invariant operators on  homogeneous Lie groups, moving to general Hörmander operators as well as heat type equa-  tions on sub-Riemannian manifolds, see [Bra14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' This parallel progression in PDE analysis  leads one to ask, how far the theory of regularity structures can be extended in these direc-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The table below gives a schematic presentation of the progress so far, presents some  open problems and places this work within the context of the study of subcritical parabolic-  type SPDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The two rows describe the two parallel progressions outlined above in the con-  text of regularity structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Translation invariant  operators on Rd  Non-translation operators on  Rd  On Riemannian manifolds  The works [Hai14, BHZ19,  BCCH21, CH16] present a  general and complete picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Mostly understood: analytic  step in [Hai14];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' aspects of  renormalisation addressed in  [BB21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  In [DDD19] 2d-PAM is treated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  A general account in  forthcoming work [HSa].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Translation invariant  operators on homogeneous  Lie groups  General Hörmander  operators  On sub-Riemannian  manifolds  Analytic theory covered in this  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Renormalisation and  convergence by hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Open problem A  Open problem B  Each problem in the second row is closely related to its equivalent problem in the first row,  we discuss the posed open problems in a little more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Open Problem A is expected to be more involved than its counterpart in the Euclidean  setting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' going from translation invariant operators to non-translation invariant oper-  ators by local approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In the case of general Hörmander operators, the un-  derlying Lie group structure would in general vary from point to point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' An interesting  application would be the study of general hypoelliptic Anderson models (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5) where  the underling particles perform a general diffusion with generator of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3), c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Open Problem B is motivated by the fact that analogous to the tangent space being an  appropriate local approximations of a Riemannian manifold, the non-holonomic tan-  gent space is an appropriate local approximations of a sub-Riemannian manifold and  7  each fibre of the non-holonomic tangent space is a stratified Lie group, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' [ABB20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Note that in view of Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3 the difficulties in Problem B are expected to be some-  what complementary to those of Problem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  NOTATION:  Given α ∈ R we let [α] := max{r ∈ Z : r ≤ α} denote the integer part of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We  often write ≲ to mean that an inequality holds up to multiplication by a constant which may  change from line to line but is uniform over any stated quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In general we reserve the  notation D for the usual derivative on Euclidean space, and X , Y for elements of a Lie algebra  thought of as vector fields on the associated Lie group, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We will use | · | to denote  the size of various quantities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' elements of a Lie group, the Haar measure of subsets of the  group, the homogeneity of members of the structure space in a regularity structure, traces  of linear maps and the absolute value function on R etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The meaning will usually be clear  from the context but in cases where it is not we will make sure to specify in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For  integrals we will use the standard notation, dx, for integration against the Haar measure on a  given homogeneous, Lie group, see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Throughout most of the article we take  an intrinsic point of view and do not equip the homogeneous Lie group with an explicit chart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  STRUCTURE OF THE PAPER:  We begin with a discussion of analysis on homogeneous Lie  groups sufficient for our purpose, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The most important result in this section is The-  orem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='13 which provides us with an intrinsic version of Taylor’s theorem on homogeneous  Lie groups and will be used frequently throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Section 3 contains the bulk of the paper  and establishes an extension to the analytic aspects of regularity structures to the setting of  homogeneous Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In Section 4 we demonstrate the application of this general theory  to semi-linear evolution equations of the type discussed in the introduction and provide an  example fixed point theorem for such generalised multiplicative stochastic heat equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  We note that there is no difficulty in extending the general fixed point theorem of [Hai14] to  our setting, we only specialise to aid the clarity of presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Finally, in Sections 5 and 6  we the construct suitable regularity structures for our specialised setting and demonstrate  a solution theory for Anderson-type equations associated to sub-Laplacians on quotients of  stratified Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The very last sections draw heavily on the notation and arguments of  [Hai14, HP15, BHZ19], which carry over to homogeneous Lie Groups to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  The authors wish to thank Martin Hairer, Rhys Steele, Ilya Chevyrev  and Ajay Chandra for helpful and insightful conversations during the preparation of this  manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  AM wishes to thank the INI and DPMMS for their support and hospitality which was sup-  ported by Simons Foundation (award ID 316017) and by Engineering and Physical Sciences  Research Council (EPSRC) grant number EP/R014604/1 as well as DFG Research Unit FOR2402  for ongoing support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  HS acknowledges funding by the Imperial College London President’s PhD Scholarship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' He  wishes to thank Josef Teichmann and Máté Gerencsér for their hospitality during his visit to  ETH Zürich and TU Wien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  8  2 ANALYSIS ON HOMOGENEOUS LIE GROUPS  We collect some basic facts on homogeneous Lie groups and smooth function on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Let  g be a Lie algebra and G be the unique, up to Lie algebra isomorphism, corresponding sim-  ply connected Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We write [·,·] for its Lie bracket and inductively set, g(1) = g and  g(n) = [g(n−1),g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Recalling the exponential map exp : g → G, we have the Baker–Campbell–  Hausdorff formula  exp(X )exp(Y ) = exp(H(X ,Y )) ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1)  where H is given by H(X ,Y ) = X + Y + 1  2[X ,Y ] + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' with the remaining terms consisting of  higher order iterated commutators of X and Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' crucially H is universal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' it does not depend  on the underlying Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 ([FS82, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Assume that g is nilpotent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' g(n) = 0 for some n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Then, the exponential map is a diffeomorphism and  through this identification of g with G, the map  G×G ∋ (x, y) �→ xy ∈ G  becomes a polynomial map (between vector spaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  the pull-back to G of the Lebesgue measure on g is a bi-invariant Haar measure on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' A dilation on g is a group of algebra automorphisms {Dr}r>0 of the form Dr X = exp(logr ·s)X  with s : g → g being a diagonalisable linear operator with 1 as its smallest eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The requirement that the smallest eigenvalue of s be 1 is purely cosmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Oth-  erwise, denoting the smallest eigenvalue by s1, one can work with the new operator ˜s = 1  s1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For a ∈ R, denote by Wa ⊂ g the eigenspace of s with eigenvalue a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Thus for  X ∈ Wa, Y ∈ Wb one has the condition  Dr [X ,Y ] = [Dr X ,Dr Y ] = r a+b[X ,Y ]  which in particular implies [Wa,Wb] ⊂ Wa+b and since Wa = {0} for a < 1  g(j) ⊂  �  a≥j  Wa .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  In particular, if g admits a family of dilations, it is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The converse is not necessarily  true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' A homogeneous Lie group G is a simply connected, connected Lie group where  its Lie algebra g is endowed with a family of dilations {Dr }r>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For r > 0, we define the group  automorphism  x �→ r · x := exp◦Dr ◦exp−1 x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  A homogeneous norm on G is a continuous function |·| : G → [0,∞) satisfying the following  properties for all x ∈ G, r ∈ R  9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' |x| = 0 if and only if x = e, the neutral element,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' |x| = |x−1| ,  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' |r · x| = r|x| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  The homogeneous norm naturally induces a topology generated by the open sets and in  turn a Borel σ-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' From now on, we will always assume that G is equipped with this  topology and σ-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Furthermore, given a homogeneous group G we denote by {X j}d  j=1 ∈  g a basis of eigenvectors of s with eigenvalues 1 = s1 ≤ s2 ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' ≤ sd and such that  sX j = sj X j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2)  Given a measurable subset E ⊂ G we write |E| for its Haar measure which we assume to be  normalized such that the set B1 =: {x ∈ G : |x| ≤ 1} has measure 1, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In  integrals we use the standard notation dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We define |s| := trace(s) as the homogeneous di-  mension of G, since for any measurable subset E ⊂ G and r > 0, one has  |r ·E| = r |s||E| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  We also define balls of radius r > 0,  Br(x) := {y ∈ G : |x−1y| < r} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  The topology induced by these balls agrees with the topology of G as a Lie group, see [FR16,  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Note that due to the non-commutativity of G, in general |x−1y| ̸= |yx−1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We  consistently work with the following choice a semi-metric on the group,  dG(x, y) := |x−1y| = |y−1x| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  For K ⊂ G we write ¯K := {z ∈ G : dG(z,K) := infy∈K|y−1z| ≤ 1} for the 1-fattening of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' A function f : G → R is called homogeneous of degree λ ∈ R, if f (r · x) = r λf (x)  for all x ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' One can show, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' [FS82, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5 & Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6], that for any homogeneous norm  on G there exists γ > 0 such that  |xy| ≤ γ(|x|+|y|) for any x, y ∈ G  ||xy|−|x|| ≤ γ|y| for any x, y ∈ G such that |y| ≤ 1  2|x| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Furthermore all homogeneous norms are mutually equivalent and we may always choose a  homogeneous norm that is smooth away from e ∈ G, [FR16, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 DERIVATIVES AND POLYNOMIALS  We identify g with the left invariant vector fields gL on G and write gR for the right invariant  vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We write Xi for the the basis elements as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2) seen as elements of gL and Yi  for the basis of gR satisfying Yi|e = Xi|e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Thus we can write  X j f (y) = ∂t f (y exp(t X j))|t=0  and  Yj f (y) = ∂t f (exp(tYj)y)|t=0  10  for any smooth function f ∈C ∞(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  A map P : G → R is called a polynomial if P ◦ exp : g → R is a polynomial on g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 Let ζi be  the basis dual to the basis Xi of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We set η j = ζj ◦ exp−1, which maps G to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Note that  η = (η1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=',ηd) forms a global coordinate system 2 and furthermore any polynomial map on G  can be written in terms of coefficients aI ∈ R as  P =  �  I  aIηI  with the sum running over a finite subset of Nd and where for a multi-index I = (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=',id) ∈ Nd  we write ηI = ηi1  1 ·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='·ηid  d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Define d(I) = �  j sji j and |I| = �  j i j, we call max{d(I) : aI ̸= 0} the  homogeneous degree and max{|I| : aI ̸= 0} the isotropic degree of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For a > 0, we denote  by Pa the space of polynomials of homogeneous degree strictly less than a and define △ =  {d(I) ∈ R : I ∈ Nd}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 3 We can rewrite the group law on G explicitly in terms of η = (η1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=',ηd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For j ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=',d} and multi-indices I, J s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' d(I) + d(J) = sj, there exist con-  stants C I,J  j  > 0 such that the following formula holds,  η j(xy) = η j(x)+η j(y)+  �  I,J̸=0, d(I)+d(J)=sj  C I,J  j ηI (x)ηJ(y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' By the Baker–Campbell–Hausdorff formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1) one has  η j(xy) = η j(x)+η j(y)+  �  |I|+|J|≥2  C I,J  j ηI (x)ηJ(y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  By setting either x = e or y = e, we find that C I,J  j  = 0 if I = 0 or J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Since furthermore  η j((r x)(r y)) = r sj η j(xy) the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5 implies that for sj < 2 one has η j(xy) = η j(x) + η j(y) while for  sj = 2 one has η j(xy) = η j(x)+η j (y)+�  sk=sl=1C k,l  j ηk(x)ηl(y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' It is also noteworthy to realise that Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5 implies that Pa is invariant  under right and left-translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' (This is not true if one replaces homogeneous degree by  isotropic degree, except if G is abelian or a = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' If follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5 that one can write  ηK (xy) = ηK (x)+ηK (y)+  �  I,J̸=0, d(I)+d(J)=d(K )  C I,J  K ηI(x)ηJ(y) ,  where the constants C I,J  K  can be written in terms of the constants C I,J  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  1Recall that the space of polynomial functions on g is canonically isomorphic to �  n(g∗)⊗sn where ⊗s denotes  the symmetric tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  2Occasionally we use the corresponding notation  ∂  ∂ηi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  3We point out a possibly counter-intuitive quirk of our definition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' for k ∈ △ the set Pk does not contain polyno-  mials of degree k but only those of degree less than k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  11  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For i, j ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=',d},  Xiη j = δi,j +  �  I̸=0, d(I)=sj −si  C I,ei  j  ηI ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3)  where ei denotes the multi-index (0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=',0,1,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=',0) with the 1 being in the i-th slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' This follows directly from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5, applying Xi to the function  y �→ η j(xy) ,  evaluating at y = 0 and using the fact that Xi|0 =  ∂  ∂ηi |0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='10 ([FS82, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' One has X j = �P j,k  �  ∂  ∂ηk  �  where  P j,k =  � 1  if k = j  0  if sk ≤ sj,k ̸= j  and P j,k is a homogeneous polynomial of degree sk −sj if sk > sj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The analogous statement  holds for the vector fields Yj,  For a multi-index I = (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=',id) ∈ Nd we introduce the notation X I = X i1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='X id  d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Note that the  order of the composition matters since g is not in general Abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' It is a well known fact that  any left invariant differential operator on G can (uniquely) be written as a linear combination  of {X I}I∈Nd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The next proposition follows as a direct consequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='11 ([FS82, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The following maps from Pa → RdimPa are linear iso-  morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' P �→  ��  ∂  ∂η  �I  P(e)  �  d(I)<a  ,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' P �→  �  X IP(e)  �  d(I)<a ,  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' P �→  �  Y IP(e)  �  d(I)<a ,  The same holds replacing e ∈ G with any other point x ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For a smooth function f : G → R, a point x ∈ G and a ∈ (0,∞), we define the  left Taylor polynomial of homogeneous degree (less than) a of f at x to be the unique polyno-  mial Pa  x[f ] ∈ Pa such that X I Pa  x[f ](e) = X I f (x) for all I such that d(I) < a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The right Taylor  Polynomial can be defined similarly, replacing X I by Y I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' One observes that for I ∈ Nd and a > d(I) one has that  X I Pa  x[f ] = Pa−d(I)  x  [X I f ]  for all f ∈ C ∞(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Indeed, this follows from the fact that for any d(J) < a − d(I) one has  X J(X I Pa  x[f ])(e) = X J(X I f )(x), since X J X I can be written as a linear combination of {X K }d(K )≤a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  12  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='13 (Taylor’s Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For each a ≥ 0 and every f ∈ C ∞(G) it holds that  f (xy)−Pa  x[f ](y) =  �  |I|≤[a]+1,d(I)≥a  �  G  X I f (xz)QI(y,dz) ,  where for each multi-index I and y ∈ G the measure QI(y, ·) is supported on Bβ[a]+1|y|(e) for  some β > 0 depending only on G and satisfies  �  G |QI(y,dz)| ≲ |y|d(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Note that, while it is very useful to have a rather explicit form of the reminder in  order establish Schauder estimates in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5, there is a more common version of Taylor’s  Theorem on homogeneous groups, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' [FS82, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='37], which states that the remainder  satisfies the estimate  |f (xy)−Pa  x[f ](y)| ≲a  �  |I|≤[a]+1,d(I)≥a  |y|d(I)  sup  |z|≤β[a]+1|y|  |X I f (xz)|  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4)  and which follows as a trivial consequence of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Furthermore the analogous  claim for the right Taylor polynomial, (PR)a  x[f ], holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In this case the analogue of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4) reads  |f (yx)−(PR)a  x[f ](y)| ≲a  �  |I|≤[a]+1,d(I)≥a  |y|d(I)  sup  |z|≤β[a]+1|y|  |Y I f (zx)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5)  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' If we define ˜Pa  x[f ](y) := Pa  x[f ](x−1y), then it follows that  f (y)− ˜Pa  x[f ](y) =  �  |I|≤[a]+1,d(I)≥a  �  G  X I f (xz)QI(x−1y,dz)  and in particular that  |f (y)− ˜Pa  x[f ](y)| ≲a  �  |I|≤[a]+1  d(I)≥α  |x−1y|d(I)  sup  |z|≤C|x−1y|  |X I f (xz)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  In the particular case when f is compactly supported, one can rewrite this for the rescaled  function f λ(x) =  1  λ|s| f (λ· x) as  |f λ(y)− ˜Pa  x[f λ](y)| ≲a  �  |I|≤[a]+1  d(I)≥a  λ−d(I)−|s||x−1y|d(I) sup  z∈G  |X I f (z)|  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6)  ≲a,f  �  δ≥0  λ−(a+δ)−|s||x−1y|a+δ ,  where the sum over δ runs over some finite subset of (0,∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Given f ∈ C ∞(G), if we define F ∈ C ∞(G × G) by setting F(y,z) = f (z−1y), we  find that  ˜Pa  x[F(·,z)](y) = ˜Pa  z−1x[f ](z−1y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For any p ∈ Pγ one has  ˜Pγ  x[p](y) = p(y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  13  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='13, given below, we use the following elementary  observations  The map  Φ : Rn → G,  (t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=',tn) �→ exp(t1X1)·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='·exp(tnXn)  is a diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' To see this note that it is clearly a local diffeomorphism since one  has  DΦ|0 : T Rd|0 ∼ Rd → T G|0 ∼ g  (t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=',tn) �→  �  i  ti Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Then since,  Φ(r s1t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=',r sn tn) = r ·Φ(t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=',tn)  it is seen to be a global diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Setting for t ∈ Rd,  |t|s :=  d�  i=1  |ti|1/si ,  one finds that there exists some β = β(G) > 0 such that  1  β|t|s ≲ |Φ(t)| ≲ β|t|s  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='7)  uniformly over t ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  By repeated use of the commutator, for any I, J ∈ Nd and i ∈ N there exist coefficients  λi,J,I such that  Xi X J =  �  I  λi,J,I X I  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='8)  and one has λi,J,I = 0 whenever d(J)+di ̸= d(I) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  If a ∈ △, then ma := max{|I| : d(I) ≤ a} = [a] where [a] denotes the integer part of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  This follows from the fact that d(I) ≤ a implies ma ≤ [a] and since ([a],0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=',0) is in the  set over which the maximum is taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Define for i ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=',d} the following measure on Rd with support on  Bs  |t|s(0)  ˜Qei (t,ds) =  �  j<i  δt j (ds j)·1[0,ti](dsi)  �  j>i  δ0(ds j)  and define Qei (y, ·) = Φ∗ ˜Qei (Φ−1(y), ·) to be the push-forward measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' First we show that,  f (xy)− f (x) =  n�  i=1  �  G  (Xi f )(xz)Qei (y,dz) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='9)  14  Indeed one can write for y = y1y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' yd, where yi = exp(ti Xi) and t = (t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=',td) = Φ−1(y)  f (xy)− f (x) =  d�  i=1  f (xy1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' yi−1yi)− f (xy1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='yi−1)  =  d�  i=1  �ti  0  ∂s f (xy1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' yi−1exp(sXi))|s=s′ds′  =  d�  i=1  �ti  0  (Xi f )(xy1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' yi−1exp(sXi))ds  =  d�  i=1  �  Rn(Xi f )(xΦ(s)) ˜Qei (t,ds)  =  d�  i=1  �  G  (Xi f )(xz)Qei (y,dz) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Using the fact that Φ is a diffeomorphism one easily checks that that Qei (y,dz) is supported  on Bβ|y|(0) where β is as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='7), and satisfies  �  G |Qei |(y,dz) ≲ |y|di , thus we have proved the  theorem in the special case a ∈ (0,1], also known as mean-value theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We turn to the  proof in the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Set g(y) = f (xy)−Pa  x[f ](y), we note that X J g(0) = 0 for d(J) < a while X J g(y) =  X J f (xy) for d(J) ≥ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We shall prove that for any multi-index J it holds that one can write  X J g(y) =  �  |I|≤[a]+1,d(I)≥a  �  G  X I f (xz)QI,J(y,dz) ,  where the measures QI,J(x,·) satisfy the following properties  QI,J(x,·) is supported on Bβm(J)|y|(0) ⊂ G where m(J) = min{m ∈ N : a −m < d(J)} , �  G |QI,J|(x,d y) ≲ |x|d(I)−d(J) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  We shall prove by induction on m ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=',⌈a⌉}, that for multi-indices J satisfying a − m <  d(J) ≤ a the claim holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  The case m = 1 follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='8) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='9),  X J g(y) = X J g(y)− X J g(0) =  d�  i=1  �  G  (Xi X J g)(z)Qei (y,dz)  =  n�  i=1  �  G  �  I : d(I)=d(J)+si  λi,J,I (X I g)(z)Qei (y,dz)  =  n�  i=1  �  G  �  I : d(I)=d(J)+si  λi,J,I (X I f )(xz)Qei (y,dz)  =  �  |I|≤[a]+1,d(I)≥a  �  G  X I f (xz)QI,J(y,dz)  The properties of QI,J(y,dz) follow from the corresponding properties of Qei (y,dz),  completing the case m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  15  Suppose the claim holds for m −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Then by the same argument as above  X J g(y) =  n�  i=1  �  d(K )=d(J)+si  λi,J,K  �  G  (X K g)(xz)Qei (y,dz)  =  �  i,K :d(K )=d(J)+si <a  λi,J,K  �  G  (X K g)(xz)Qei (y,dz)  +  �  i,K :d(K )=d(J)+si ≥a  λi,J,K  �  G  (X K g)(xz)Qei (y,dz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  We can apply the induction hypothesis to the terms in the first sum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' since d(K ) = d(J)+  si ≥ d(J)+1 ≥ a −(m −1) and find  (X K g)(xz) =  �  |I|≤[a]+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='d(I)≥a  �  G  X I f (x ˜z)QI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K (z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='d ˜z)  and therefore  X J g(y) =  �  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K :d(K )=d(J)+si <a  �  G  (X K g)(xz)Qei (y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='dz)  +  �  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K :d(K )=d(J)+si ≥a  λi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K  �  G  (X K g)(xz)Qei (y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='dz)  =  �  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K :d(K )=d(J)+si <a  λi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K  �  |I|≤k+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='d(I)≥a  �  G  �  G  X I f (x ˜z)QI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K (z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='d ˜z)Qei (y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='dz)  +  �  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K :d(K )=d(J)+si ≥a  λi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K  �  G  (X K g)(xz)Qei (y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='dz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  =  �  i,K :d(K )=d(J)+si <a  λi,J,K  �  |I|≤k+1,d(I)≥a  �  G  X I f (x ˜z)  �  QI,K ∗Qei �  (y,d ˜z)  +  �  i,K :d(K )=d(J)+si ≥a  λi,J,K  �  G  (X K g)(xz)Qei (y,dz)  =  �  |I|≤[a]+1,d(I)≥a  �  G  X I f (x ˜z)QI,J(y,d ˜z)  whereQI,J is defined such that the last line holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Concerning the measuresQI,J(y,·),  we note that  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' suppQI,J(y,·) ⊂ Bβm(J)|y| since  supp  �  QI,K ∗Qei (y,·)  �  ⊂ Bβm(K )+1|y|  as suppQI,K (y,·) ⊂ Bβm(K )|y|(0) and suppQei (y,dz) ⊂ Bβ|y|(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  �  G |QI,J(y,·)| ≲ |x|d(I)−d(J) since  �  |QI,K ∗Qei |(y,dz) ≤  �  |QI,K |(y,dz) ·  �  |Qei |(y,dz) ≲ |y|d(I)−d(K ) ·|y|di .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  16  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2 DISTRIBUTIONS AND CONVOLUTION  We define D(G) := C ∞  c (G) to be the space of compactly supported smooth functions on G  equipped with the natural Fréchet topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The space of distributions on G is given by the  dual D′(G) of D(G) and for ξ ∈ D′(G) and φ ∈ D(G) we either write 〈φ,ξ〉 or ξ(φ) for the canon-  ical pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Given r ∈ R≥0 we introduce the following useful space of test functions  Bλ  r (x) :=  �  φ ∈C ∞  c (Bλ(x)) : |X Iφ| ≤  1  λ|s|+d(I) ∀I : d(I) ≤ r  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  We will often use the shorthand Br := B1  r (e) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Given φ ∈ D(G) we extensively use the following  notations,  φλ  x(z) := 1  λ|s| φ  � 1  λ ·(x−1z)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Recalling that we use the notation dx for the Haar measure on G we define the norms,  ∥f ∥Lp :=  ���  G |f (x)|p dx  �1/p ,  for 1 ≤ p < ∞,  esssupx∈G |f (x)|,  p = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  By left and right translation invariance of the Haar measure, for f ∈ L1(G) it holds that  �  G  f (yx)dy =  �  G  f (xy)dy =  �  G  f (y)dy =  �  G  f (y−1)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  See the discussion after [FR16, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' From the scaling properties of D it follows that  Bλ  r (x) =  �  φλ  x : φ ∈ B1  r (e)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  One defines the convolution of two functions φ,ψ : G → R as  ψ∗φ(x) =  �  ψ(y)φ(y−1x)d y =  �  ψ(xy−1)φ(y)d y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='10)  Note that in general ψ∗φ(x) ̸= φ∗ψ(x) but still many of the usual inequalities hold, in par-  ticular the Young inequity, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' [FS82, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' One can write  ψ∗φ(x) =  �  ψ(y)φy(x)d y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  We extend convolution to generalised function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' elements of D′(G), by duality in the usual  manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' From now on we will use the notation ˜φ(z) := φ(z−1), one notes that the following  identities hold  〈f ,g ∗φ〉 = 〈 ˜g∗f ,φ〉 = 〈f ∗ ˜φ,g〉  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='11)  �  f ∗ g = ˜g ∗ ˜f  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='12)  ψλ ∗φλ = (ψ∗φ)λ  and  (ψ∗φ)x = ψx ∗φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  17  Recalling the notation X I = X i1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='X id  d for any multi-index I = (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=',id) ∈ Nd, we introduce the  analogous notation Y I = Y id  d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='.Y i1  1 for the right invariant basis vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Note the reversal in  the composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Since for any right invariant vector field Y and left invariant vector field  X such that X |e = Y |e it holds that 〈X f ,g〉 = 〈f ,Y g〉 for f , g ∈ C ∞  c (G), it follows from the  definition of the convolution that  X I(ψ∗φ) = ψ∗(X I φ),  Y I(ψ∗φ) = (Y Iψ)∗φ,  (X Iψ)∗φ = ψ∗Y Iφ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For α ∈ R we define the space Cα(G) of α-Holder continuous distributions to be  those f ∈ D′(G) such for every compact set K ⊂ G,  If α ≤ 0  ∥f ∥Cα(K) = sup  x∈K  sup  φ∈B⌈−α⌉  sup  λ∈(0,1)  |〈f ,φλ  x〉|  λα  < +∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  If α > 0, there exists a continuous map G → Pα, x �→ ˜Px such that  ∥f ∥Cα(K) = sup  x∈K  sup  φ∈B0  sup  λ∈(0,1)  |〈f − ˜Px,φλ  x〉|  λα  +sup  x∈K  | ˜Px|Pα < +∞ ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='13)  where | · |Pα denotes any norm on the finite dimensional vector space Pα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Furthermore,C(G) denotes the space of continuous functions (note that only the strict inclusion  C(G) ⊊ C0(G) holds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We note that as in [Hai14], for α ∈ △ the Cα(G) spaces do not agree with the  usual notion of continuously α-differentiable functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' This definition, however, is more  canonical in the context of regularity structures, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  The following proposition gives some useful properties that also mirror those of Euclidean  Hölder spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Given any real number α ∈ R the following holds  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Cα(G) ⊂ Cβ(G) for every β < α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For any multiindex I the map X I : Cα(G) → Cα−d(I)(G),  f �→ X I f is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' If α > 0, then any distribution f ∈ Cα(G) agrees4 with a continuous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Note that this proposition in particular implies that C ∞(G) = ∩α>0Cα(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  4As usual, we say a distribution agrees with a continuous functions, if it lies in the image of the canonical em-  bedding C(G) → D′(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  18  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Points 1 and 2 follow directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For Point 3, denote by ˜Px the polynomial in Defini-  tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4, we set F = f − ˜P·(·), then for any ψ ∈ B0 such that  �  G ψ = c−1 > 0 and smooth com-  pactly supported function g we find  〈F,g〉 = c lim  λ→0〈F,g ∗ψλ〉 = c lim  λ→0  �  F,  �  G  g(y)ψλ  yd y  �  = c lim  λ→0  �  G  〈F,ψλ  y〉g(y)d y = 0 ,  where in the last step we used that since y → ˜Py is continuous we have  〈F,ψλ  y〉 = 〈f − ˜Py(·),ψλ  y〉+  �� ˜Py(z)− ˜Pz(z)  �  ψλ  y (z)dz → 0 ,  as λ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Thus we find that the distribution f agrees with the continuous function x �→ ˜Px(x),  concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Note that by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='22 for α > 0 and any f ∈ Cα(G) we know that X I f  agrees with a continuous function whenever d(I) < α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Thus, in particular Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3 of  Taylor polynomials Pα  x [f ] extends to Cα(G) ⊃C ∞(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3 DISCRETE SUBGROUPS  Recall that, given a discrete subgroup G ⊂ G acting on G (say) on the left, then the quotient  space S := G/G = {Gx : x ∈ G} is a smooth manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Furthermore, the quotient map π : G → S  is a smooth normal covering map , c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' [Lee13, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='29] and the canonical G right action  S ×G → S,  (Gx,x′) �→ Gxx′  makes it a homogeneous space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We call G a lattice if S = G/G is a compact space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' This is  equivalent to S carrying a finite G invariant measure, which we shall denote by d(Gx), c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  [Rag07, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Throughout this article we assume that every homogeneous Lie group we  work with carries a lattice, the following theorem, [Rag07, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='12], gives sufficient condi-  tions for this to be the case, which all our examples satisfy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Let G be a simply connected nilpotent Lie group and g its Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Then, G  admits a lattice if and only if g admits a basis with respect to which the structure constants of  g are rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Let us point out that there do exist nilpotent Lie groups that do not admit a basis with re-  spect to which the structure constants are rational, see [Rag07, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='14] for an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Denote by π∗ : C(S) →C(G) the pull-back under π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We define a convolution map  ∗S : C ∞(S)×C ∞  c (G) →C ∞(S),  (f ,φ) �→ f ∗S φ  as the unique map, such that the following diagram commutes  C ∞(S)×C ∞  c (G)  C ∞(S)  C ∞(G)×C ∞  c (G)  C ∞(G) ,  ∗S  π∗×id  π∗  ∗  19  where convolution on the bottom row is convolution on the group as defined in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In order  to check that this map is well defined, we use the following terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' A function f ∈C ∞(G)  is called left G periodic, if for every x ∈ G and n ∈ G it holds that  f (x) = f (nx) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Thus, we only need to check that for any φ ∈ C ∞  c (G) and any left G periodic function f ∈  C ∞(G), the function f ∗φ is left G periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Indeed for any n ∈ G and x ∈ G, it holds that  f ∗φ(nx) =  �  G  f (y)φ(y−1nx)d y =  �  G  f (y)φ((n−1y)−1x)d y =  �  G  f (ny)φ(y−1x)d y =  �  G  f (y)φ(y−1x)d y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  By duality, one naturally extends the notion of convolution on S to pairs, (ξ,ζ) ∈ D′(S)×D′  c(G),  where D′  c(G) denotes distributions on G with compact support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4 CONCRETE EXAMPLES  In order to cement ideas we present two concrete examples of non-abelian, homogeneous  Lie groups, both of which are identified with fixed global charts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' These will be revisited in  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4 when we discuss the heat operator on the Heisenberg group and Kolmogorov  type operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 THE HEISENBERG GROUP  Given n ≥ 1 we equip define the Heisenberg group Hn as the set R2n ×R with the group law,  (x, y,z)(x′, y′,z′) =  �  x + x′, y + y′,z + z′ +  n�  i=1  �  x′  i yi − xi y′  i  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Note that one may equally define the complex Heisenberg group on Cn × R  equipped with the group law,  (u,z)(u′,z′) =  �  v + v′,z + z′ +ℑ  n�  i=1  ui ¯u′  i  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  One sees that these definitions are equivalent after identifying Cn with R2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  The origin e = (0,0,0) is clearly the identity and for (x, y,z) ∈ R2n × R one has (x, y,z)−1 =  (−x,−y,−z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The Lie algebra, h of Hn is identified with R2n+1 and spanned by the basis of  left-invariant vector fields,  Ai(x, y,z) = ∂xi + yi∂z,  Bi(x, y,z) = ∂yi − xi∂z,  C(x, y,z) = ∂z  and equipped with the Lie bracket [A,B] = AB −B A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We observe that for any (x, y,z) ∈ Hn and  i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=',n,  [Ai,Bi](x, y,z) = −2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='14)  We say that a Lie algebra is graded if there exist vector spaces {Wk}k≥1 where only finitely  many Wk are non-zero, g = �∞  k=1Wk and [Wi,Wj] ⊂ Wi+j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  20  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' A homogeneous Lie group G is called stratified (or a Carnot group) if its Lie  algebra is graded and generated by W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Due to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='14), we see that if we equip Hn with the dilation map,  λ·(x, y,z) := (λx,λy,λ2z),  then Hn becomes a stratified group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Refer to [Bra14, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6] for a discussion of generalisa-  tions of this structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  A simple example of a lattice on the Heisenberg group is the set of integer vectors (a,b,c) ∈  Hn := Z2n × Z equipped with the group law as defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Note that it is not a normal  subgroup and thus the quotient space Hn/Hn not a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' However, since it is a lattice the  homogeneous space Hn/Hn is compact, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2 MATRIX EXPONENTIAL GROUPS  For n ≥ 1, let B be a rational n ×n block matrix of the form,  B =  \uf8eb  \uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed  0  B1  0  ···  0  0  0  B2  ···  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  0  0  ···  0  Bk  0  0  ···  0  0  \uf8f6  \uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8  with each Bi a pi−1×pi block matrix of rank pi, where n ≥ p0 ≥ p1 ≥ ··· ≥ pk and �k  i=0 pi = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Note that this implies the zero blocks on the diagonal are all pi × pi square matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We can  equip R×Rn with an associated Lie structure by defining the group law  (t,z)(s,z′) :=  �  t + s,z′ +exp  �  sB⊤�  z  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  To define the dilation we begin by decomposing according to the structure of the block matrix  B, writing  Rn = Rp0 ×···×Rpn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Thus the action of B⊤ on z = (z0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=',zk) ∈ Rp0×···×Rpk is written B⊤z = (B⊤  1 z0, B⊤  2 z1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='B⊤  k zk−1)  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Then we set  λ·(t,z0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=',zk) := (λ2t,λz0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=',λ2k+1zk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  The origin e = (0,0) is again the identity element and (t,z)−1 = (−t,−exp(−tB⊤)z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The Lie  algebra is spanned by the translation invariant vector fields,  Xi(t,z) = ∂zi  for i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=',p0  and  Y (t,z) = ∂t −(Bz)·∇ = ∂t −  n�  i,j=p0+1  bi j zi∂zi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  This defines the matrix exponential group associated to B and one sees that it is not stratified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  The simplest non-trivial example is to set n = 2 and  B =  �0  1  0  0  �  ,  21  so that using the suggestive notation (t,v,x) ∈ R×R×R, the group law becomes,  (t,v,x)(s,w, y)= (t + s,v + w,x + y + sv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  The equivalent scaling as above is to set λ · (t,x,v) = (λ2t,λv,λ3x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We refer to [Man97] for  more details and Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4 below for a discussion of natural, second order, hypoelliptic,  linear operators associated to these groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  3 REGULARITY STRUCTURES AND MODELS  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' A regularity structure is a pair T = (T,G) consisting of the following elements:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' A graded vector space T = �  α∈A Tα where  the index set A ⊂ R is discrete, bounded from below and contains zero,  each Tα is finite dimensional with a fixed norm |·|α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We write Qα : T → Tα for the  canonical projection,  T0 is isomorphic to R with a distinguished element 1 ∈ T0, such that |1|0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  The space T is called the structure space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' A group G of linear operators acting on T, such that for every Γ ∈ G it holds that Γ|T0 is  the identity map and for all τ ∈ Tα:  Γτ−τ ∈  �  β<α  Tβ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  The group G is called the structure group of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  A sector is a G invariant subspace V ⊂ T and if V ̸= {0} the regularity of the sector V is defined  as min{α ∈ A : V ∩Tα ̸= {0}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  We make use of the natural shorthands, T>α, T≥α, T<α, T≤α and the projections Q>α etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Given a regularity structureT = (T,G) and r ∈ N such that r > |min A|, a model  for T is a pair M = (Π,Γ), consisting of  a realisation map Π : Rd → L(T,D′(G)), x �→ Πx, such that for any compact set K ⊂ G one  has ∥Π∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K := supx∈K ∥Π∥γ,x < +∞ for all γ > 0, where  ∥Π∥γ,x :=  sup  ζ∈A∩(−∞,γ)  sup  τ∈Tζ  sup  λ<1  sup  φ∈Br  |〈Πxτ,φλ  x〉|  |τ|ζλζ  ,  a re-expansion map Γ : Rd ×Rd → G, (x, y) �→ Γx,y, which satisfiesthe algebraic condition  ΠxΓx,y = Πy  and the analytic condition ∥Γ∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K := supx,y∈K: |y−1x|<1∥Γ∥x,y,γ < +∞ for all γ > 0, where  ∥Γ∥x,y,γ :=  sup  ζ∈A∩(−∞,γ]  sup  A∋β<ζ  sup  τ∈Tζ  |Γx,yτ|β  |y−1x|ζ−β|τ|ζ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  22  Lastly, we denote by MT the space of models for T equipped with the semi-norms ∥M∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K :=  ∥Π∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K +∥Γ∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In the rest of the article, often without any further comment, we will write  r := min{n ∈ N : n > |min A|} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 THE POLYNOMIAL REGULARITY STRUCTURE  As an important example we describe the polynomial regularity structure and canonical model  which are crucial in the analysis of singular SPDEs on homogeneous Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We de-  fine the structure space ¯T to be the symmetric tensor (Hopf) algebra generated by {ηηηi}d  i=1,  which we think of as abstract lifts of the monomials {ηi}d  i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For a multi-index I, we write  ηηηI :=ηηηi1  1 ·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='·ηηηid  d as well as 1 :=ηηη0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The group ¯G is given by a copy of G acting by  g �→  �  Γg :ηηηj �→ηηηj +  �  I,J̸=0, d(I)+d(J)=sj  C I,J  j ηI(g)ηηηJ +η j (g)1  �  and ΓgηηηI =  �  Γgηηη  �I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The canonical polynomial model is defined by setting  Πxηηηj(z) = η j(x−1z)  and  Γx,yηηηj =ηηηj +  �  I,J̸=0, d(I)+d(J)=sj  C I,J  j ηI (y−1x)ηηηJ +η j(y−1x)1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  These maps are extended multiplicatively to all of ¯T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Using Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5 in the third equality,  ΠxΓx,yηηηj(z) = Πx  �  ηηηj +  �  I,J̸=0, d(I)+d(J)=dj  C I,J  j ηI (y−1x)ηηηJ +η j (y−1x)  �  (z)  = η j(x−1z)+  �  I,J̸=0, d(I)+d(J)=dj  C I,J  j ηI(y−1x)ηJ(x−1z) +η j(y−1x)  = η j(y−1xx−1z)  = η j(y−1z)  = Πyηηηj(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 DERIVATIVES AND ABSTRACT POLYNOMIALS  Next we lift the vector fields X i to abstract differential operators X i on ¯T of degree sj, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6 given later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We set  X iηηηj = δi,j1+  �  I̸=0, d(I)=sj −si  C I,ei  j  ηηηI  and extend this definition to the whole regularity structure by the Leibniz rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The two con-  ditions to be an abstract differential operator are checked directly:  23  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Indeed X i : Tα �→ Tα−si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' By the Leibinz rule the second property is checked by showing that  X iΓx,yηηηj = Γx,yX iηηηj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1)  Note that ΠxX iηηηj = XiΠxηηηj , since  ΠxX iηηηj(z) = Πx  �  δi,j1 +  �  I̸=0, d(I)=sj −si  C I,ei  j  ηηηI�  = δi,j +  �  I̸=0, d(I)=sj −si  C I,ei  j  ηI(x−1z)  and using left invariance of the vector field Xi as well as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3)  XiΠxηηηj(z) = (Xiη j(x−1·))(z) = (Xiη j)(x−1z) = δi,j +  �  I̸=0, d(I)=sj −si  C I,ei  j  ηI (x−1z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Thus (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1) follows from the injectivity of the maps Πx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In addition to showing that Xi is an abstract differential operator, we have shown  that it also realizes Xi for the polynomial model, see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6 in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2 ABSTRACT TAYLOR EXPANSIONS  The following operator, which sends a smooth function to its abstract Taylor expansion, will  be used throughout the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For a > 0 and x ∈ G, we define the family of maps  PPPa  x : Ca(G) → ¯T  where PPPa  x[f ] is characterised by the fact that ΠePPPa  x[f ] = Pa  x[f ] ∈ Pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Note that the mapPPPa  x is well defined by Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='23 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We shall almost exclusively  use it, when its argument is a smooth function f ∈C ∞(G) ⊂ Ca(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Let f ∈C ∞(G) and a ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Then for each multi-index I the coefficient of ηηηI in PPPa  x[f ]  is given by a linear combination (depending only on G) of {X K f (x)}d(K )≤d(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Furthermore, for  b ≥ a one has  PPPa  x[f ] = Q≤aPPPb  x[f ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We first observe that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2) holds since the polynomial ΠeQ≤aPPPb  x[f ] satisfies the prop-  erty required in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The first part of the lemma follows by combining Proposi-  tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='11 with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In the setting of the above lemma one has  |X IΠePPPa  x[f ](z)| = |X I Pa  x[f ](z)| ≲  �  d(I)≤d(J)<a  sup  d(J′)≤d(J)  |X J′ f (x)||z|d(J)−d(I) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Since the coefficient of ηJ in Pa  x[f ] is bounded by a universal multiple of supd(J′)≤d(J) |X J′ f (x)|  by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4 the claim follows using the fact that |X IΠeηηηJ(z)| ≲ |z|d(J)−d(I) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  24  The next lemma will be useful when proving Schauder estimates in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5, since it  allows us to circumvent the issue of having non-explicit Taylor polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Let δ > 0 and P ∈ ¯T<δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' If, for some ε ∈ [0,1],  ��(X IΠeP)(e)  �� ≤ εδ−d(I)  for all I ∈ Nd such that δ > d(I), then it holds that |P|a ≲δ,G εδ−a for all a ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We observe that Pδ  x[ΠeP] = ΠeP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Thus, the claim follows directly from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2 MODELLED DISTRIBUTIONS  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Given a regularity structure T = (T,G), a model M = (Π,Γ) and γ ∈ R we define  Dγ  M as the space of all continuous maps f : G → T<γ, such that for all ζ ∈ A ∩ (−∞,γ) the  following bounds hold for every compact set K ⊂ G  sup  x∈K  |f (x)|ζ < +∞,  sup  x,y∈K,  0<|y−1x|≤1  |f (y)−Γx,y f (x)|ζ  |y−1x|γ−ζ  < +∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  We define the corresponding semi-norm ∥ · ∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K on Dγ  M by setting,  ∥f ∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K := sup  x∈K  sup  ζ<γ  |f (x)|ζ +sup  ζ<γ  sup  x,y∈K,  0<|y−1x|≤1  |f (y)−Γx,y f (x)|ζ  |y−1x|γ−ζ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Given two models M = (Π,Γ), ¯M = ( ¯Π, ¯Γ), two modelled distributions f ∈ Dγ  M, ¯f ∈ Dγ  ¯M and a  compact set K ⊂ G we define the quantity,  ∥f ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' ¯f ∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K := sup  x∈K  sup  ζ<γ  |f (x)− ¯f (x)|ζ +  sup  (x,y)∈K  0<|y−1x|≤1  sup  ζ<γ  |f (y)− ¯f (y)−Γx,y f (y)+ ¯Γy,x ¯f (x)|ζ  |y−1x|γ−ζ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3)  For the set of modelled distributions taking values in a sector V ⊂ T we write Dγ  M(V ) and if the  regularity of the sector is α ∈ A we often use the shorthandDγ  α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We will freely drop the explicit  dependence on the model, image sector and its regularity when the context is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 RECONSTRUCTION  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='7 (Reconstruction Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Let T = (T,G) be a regularity structure with α =  min A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Then for every γ > 0 and M = (Π,Γ) ∈ MT , there exists a unique, continuous linear  map RM : Dγ  M → Cα, called the reconstruction operator associated to M, such that for any  compact K and λ ∈ (0,1]  sup  ψ∈Bm  |〈RM f −Πx f (x),ψλ  x〉| ≲K λγ∥f ∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='B2λ(x)∥Π∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='B2λ(x) ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4)  25  uniformly over x ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Furthermore the map M → RM is locally Lipschitz continuous in the  sense that for a second model ¯M = ( ¯Π, ¯Γ) and ¯f ∈ Dγ  ¯M, for every λ ∈ (0,1]  sup  ψ∈Bm  |〈RM f −R ¯M ¯f −Πx f (x)+ ¯Πx ¯f (x),ψλ  x〉| ≲K λγ�  ∥f ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' ¯f ∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='B2λ(x)∥ ¯Π∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='B2λ(x)  +∥f ∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='B2λ(x)∥Π− ¯Π∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='B2λ(x)  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5)  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Existence of a reconstruction operator for γ ≤ 0 also holds, however, uniqueness  does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In the case γ = 0 the analogous bound to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4) contains an additional logarithmic  correction on the right hand side, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' [CZ20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' It follows from the definition of the reconstruction operator, that if f ∈ Dγ  α,M is a  modelled distributions with values in a sector V ⊂ T of regularity α ≥ α, then RM f ∈ C α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  In order to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='7 we require two preliminary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' As in [FH20, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4]  we make the observation that for every N ≥ 0 there exists a ρ : G → R, smooth and compactly  supported in B1(0) and such that  �  ηI (x)ρ(x)dx = δI,0,  0 < d(I) ≤ N,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6)  where the δ here denotes the Kronecker delta applied componentwise to the multi-index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For  r > 1 we define ρ(n)(x) := rn|s|ρ(rn · x) as well as,  ρ(n,m) = ρ(n) ∗ρ(n+1) ∗···∗ρ(m),  with the convention ρ(n,n) = ρ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We then have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' If r > ∥ρ∥L1 > 1, then there exists a smooth, compactly supported function ϕ(n) =  limm→∞ ρ(n,m), where the convergence takes place in D(G) and supp(ϕ(n)) ⊂ BCr−n for C =  r  r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' First, note that since ∥ρ(m)∥L1 = ∥ρ∥L1 and ρ(m) is supported on a ball of radius r, it  follows from the mean value theorem on G (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5) with a = 0) that  |f ∗ρ(m)(x)− f (x)| =  ����  �  G  (f (y)− f (x))ρ(m)(y−1x)dy  ���� ≲ max  i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=',d ∥Yi f ∥L∞∥ρ∥L1r−m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Secondly, since  Y Iρ(n,m) = Y I(ρ(n) ∗ρ(n+1,m)) = (Y Iρ(n))∗ρ(n+1,m)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='7)  one finds by applying Young’s convolution inequality m-times, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2, that  ∥Yiρ(n,m)∥L∞ ≤ ∥Yiρ∥∞∥ρ∥m−n−1  L1  and therefore  ∥ρ(n,m) −ρ(n,m−1)��L∞ = ∥ρ(n,m−1) ∗ρ(m) −ρ(n,m−1)∥L∞  ≲ max  i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=',n∥Yiρ(n,m−1)∥L∞∥ρ∥L1r−m  ≤ ∥Y ρ∥L∞∥ρ∥m−n−2  L1  r−m  ≤ ∥Y ρ∥L∞  ∥ρ∥n+2  L1  �∥ρ∥L1  r  �m  26  which is summable in m since we assumed that r > ∥ρ∥L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Thus we may write,  ρ(n,m) = ρ(n) +  m−n−1  �  k=0  ρ(n,m−k) −ρ(n,m−1−k),  and it follows that ρ(n,m) converges uniformly as m → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='7) we obtain convergence  in D(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' It remains to check the support of ϕ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For two functions f1, f2 such that supp(fi) ∈  Bri one has supp(f1 ∗ f2) ∈ Br1+r2, hence it follows that ϕ(n) is supported in a ball of radius  �∞  m=n r−m =  r−n  1−r−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  It follows from the definitions that ϕ(n) = ρ(n) ∗ϕ(n+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We set  ˜ρ(m,n) := ˜ρ(m) ∗ ˜ρ(m−1) ∗···∗ ˜ρ(n)  and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='12) we note that ˜ϕ(m+1) ∗ ˜ρ(m) = ˜ϕ(m) and ˜ρ(m,n) → ˜ϕ(n) in D(G) as m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Let r > 1 and ρ be as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='10 Let α > 0 and ξn : G → R be a sequence of  functions such that for every compact K ⊂ G there exists a CK such that supx∈K |ξn(x)| ≤ CKrαn  and such that ξn = ξn+1 ∗ ˜ρ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Then the sequence ξn is Cauchy in C−β(G) for every β > α and  the limit ξ satisfies ξn = ξ∗ ˜ϕ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' If furthermore, for some x ∈ G and γ > −α one has the bound  |ξn(y)| ≤ rαn �  |x−1y|γ+α +r−(γ+α)n�  uniformly over n ≥ 0 and y ∈ G such that |x−1y| ≤ 1, then |〈ξ,ψλ  x〉| ≲ λγ for all λ ≤ 1 and φ ∈ Br ,  where r = −[−α].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The proof follows along the same steps as that of [FH20, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='24], but one has to  be careful since convolution is non-commutative in our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Let λ ∈ (0,1] and ψ ∈ Br , we  first establish the bound  |〈ξn −ξn+1,ψλ  x〉| ≲ λ−βr(α−β)n  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='8)  uniformly over ψ ∈ Br , λ ∈ (0,1] and locally uniformly over x ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' First observe the trivial  bound,  |〈ξn −ξn+1,ψλ  x〉| ≤ sup  x∈K  (|ξn(x)|+|��n+1(x)|)∥ψλ  x∥L1 ≤ (1+rα)CK ¯Crαn,  where ¯C := sup{  �  |ψλ(x)|dx : ψ ∈ Br } < +∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Hence, when λ ≤ r−n the bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='8) holds  directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  In the case r−n ≤ λ, using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='11) we rewrite  |〈ξn −ξn+1,ψλ  x〉| = 〈ξn+1 ∗ ˜ρ(n) −ξn+1,ψλ  x〉| = |〈ξn+1,ψλ  x ∗ρ(n) −ψλ  x〉|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  By Taylor’s theorem and in particular Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='15  |ψλ  x(z)− ˜Pr  y[ψλ  x](z)| ≲r  �  δ>0  λ−(r+δ)−|s||y−1z|r+δ ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='9)  where the sum runs over a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='10), Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='8 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6) that we have  ˜Pr  y[ψλ  x]∗ρ(n)(z) =  �  ˜Pr  y[ψλ  x](zy−1)ρ(n)(y)d y =  �  ˜P0  y[ψλ  x](zy−1)ρ(n)(y)d y = ψλ  x(y)  27  and thus  ψλ  x ∗ρ(n)(y)−ψλ  x(y) = (ψλ  x − ˜Pr  y[ψλ  x])∗ρ(n)(y),  which by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='9) is bounded uniformly by a multiple of �  δ>0 λ−(r+δ)−|s|r−n(r+δ) and supported  on a ball of radius λ+r−n ≤ 2λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Using the bound |ξn+1| ≲α rαn we conclude that  |〈ξn+1,ψλ  x ∗ρ(n) −ψλ  x〉| ≲  �  δ>0  λ−(r+δ)r−n(r+δ)rαn ≲ λ−βr(α−β)n  where we used r−n ≤ λ and without loss of generality assumed that r ≥ β in the last line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Hence {ξn}n≥1 is Cauchy in C−β(G) and for any test function,  〈ξn,ψ〉 = 〈ξn+1,ψ∗ρ(n)〉 = 〈ξm,ψ∗ρ(n,m)〉 = 〈ξ,ψ∗ϕ(n)〉,  showing that ξn = ξ∗ ˜ϕ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  To prove the second claim, for any test function ψ ∈ Br , λ > 0 and x ∈ G we write,  〈ξ,ψλ  x〉 = 〈ξn,ψλ  x〉+  �  k≥n  〈ξk+1 −ξk,ψλ  x〉,  where n is chosen so that λ ∈ [r−(n+1),r−n] and as a consequence  |〈ξn,ψλ  x〉| ≤ λ−|s|rαn  �  Bλ(x)  �  |x−1y|γ+α +r−(γ+α)n�  dy,≲ λγ+αrαn +r−γn ≲ λγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  To bound the summands 〈ξk+1 − ξk,ϕλ  x〉 we proceed as in the proof of the first claim to find  that  |〈ξk −ξk+1,ψλ  x〉| = |〈ξk+1,ψλ  x ∗ρ(k) −ψλ  x〉|  = |〈ξk+1,(ψλ  x − ˜Pr  x[ψλ  x]))∗ρ(k)〉|  ≲  �  δ>0  λ−(r+δ)−|s|r−k(r+δ)  �  B2λ(x)  |ξk+1(y)|dy  ≲  �  δ>0  λ−(r+δ)−|s|rk(α−r−δ)  ��  B2λ(x)  |x−1y|γ+αdy +r−(γ+α)(k+1)λ|s|  �  ≲  �  δ>0  �  λγ+α−r−δrk(α−r−δ) +λ−(r+δ)r−k(γ+r+δ)�  where the sum in δ is again over a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Since r + δ > α the quantity on the left is  summable over k ≥ n and is of order λγ, concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  We are now ready to proof Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For m > 0 we first define the operators R(m,m) : Dγ →C(G) by setting,  �R(m,m) f  �  (y) :=  �  Πy f (y)∗ ˜ϕ(m)�  (y) = 〈Πy f (y),ϕ(m)  y  〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  We then set, for n < m,  R(m,n) f = R(m,m) f ∗ ˜ρ(m−1,n)  28  and recalling that ˜ϕ(m+1) ∗ ˜ρ(m) = ˜ϕ(m) we find  R(m,n) f −R(m+1,n) f = R(m,m) f ∗ ˜ρ(m−1,n) −R(m+1,m+1) f ∗ ˜ρ(m,n)  =  �R(m,m) f −R(m+1,m+1) f ∗ ˜ρ(m)�  ∗ ˜ρ(m−1,n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Using the identity  (F ∗ ˜φ)(x) = 〈F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='φx〉 =  �  F(y)φx(y)d y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  it follows that  �R(m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='n) f −R(m+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='n) f  �  (x) =  ��R(m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='m) f −R(m+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='m+1) f ∗ ˜ρ(m)�  (y)ρ(m−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='n)  x  (y)d y  =  ��  Πy f (y)∗ ˜ϕ(m) −R(m+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='m+1)f ∗ ˜ρ(m)�  (y)ρ(m−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='n)  x  (y)d y  =  ���  Πy f (y)∗ ˜ϕ(m+1) −R(m+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='m+1) f  �  ∗ ˜ρ(m)�  (y)ρ(m−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='n)  x  (y)d y  =  ���  Πy f (y)∗ ˜ϕ(m+1) −R(m+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='m+1) f  �  (z)ρ(m)  y  (z)dzρ(m−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='n)  x  (y)d y  =  ���  Πy f (y)∗ ˜ϕ(m+1) −Πz f (z)∗ ˜ϕ(m+1)�  (z)ρ(m)  y  (z)dzρ(m−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='n)  x  (y)d y  =  ��  〈Πy f (y)−Πz f (z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='ϕ(m+1)  z  〉ρ(m)  y  (z)dzρ(m−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='n)  x  (y)d y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Therefore, using the fact that Πz = ΠyΓyz, we have,  �R(m,n) f −R(m+1,n) f  �  (x) =  ��  〈Πy(f (y)−Γyz f (z)),ϕ(m+1)  z  〉ρ(m)  y  (z)dzρ(m−1,n)  x  (y)d y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Then successively applying the facts that,  supy,z∈Br−m (0) |〈Πyτ,ϕ(m+1)  z  | ≲ ∥Π∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='Br−m(0)r−αm|τ|ζ for τ ∈ Tζ,  ∥f (y)−Γyz f (z)∥α ≲ ∥f ∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='Br−m (y)r(α−γ)m uniformly over |y−1z| ≲ r−m  ∥ρ(n,m−1)  x  ∥L1 ≲ 1 uniformly over m > n ≥ 0,  we establish the bound,  ∥R(m,n) f −R(m+1,n) f ∥L∞(K) ≲ ∥Π∥γ, ¯K∥f ∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' ¯Kr−γm ,  uniformly over m ≥ n ≥ 0, where ¯K denotes the two fattening of the set K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' It follows directly  from the definition and properties of a model that we also have the bound,  ∥R(n,n) f ∥L∞(K) ≲ ∥Π∥γ, ¯K∥f ∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' ¯Kr−αn,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='10)  where α = min A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' It follows that R(m,n) f converges uniformly on compacts as m → ∞ to  R(n) f which also satisfies the bound(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Since it also holds that for every m ≥ n +1,  R(m,n) f = R(m,m) f ∗ ˜ρ(m−1,n) = R(m,m) f ∗ ˜ρ(m−1,n+1) ∗ ˜ρ(n) = R(m,n+1) f ∗ ˜ρ(n) ,  29  we find that  R(n) f = R(n+1) f ∗ ˜ρ(n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Therefore we may apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='11 to see that there exists a limit Rf := limn→∞ R(n) f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  With validity of the limit established we now turn to show the bounds (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' this  requires us to keep more careful track of the underlying sets in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We begin with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4),  first noting that if we define fx(y) := Γy,x f (x) then one has R(n,n) fx = Πx f (x) ∗ ˜ϕ(n) so that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4) can be written as the claim that for all λ ∈ (0,1],  sup  ψ∈Bm  |〈R(f − fx),ψλ  x〉| ≲K λγ∥f ∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='B2λ(x)∥Π∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='B2λ(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='11)  Using that |(f − fx)(z)|α ≲ ∥f ∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='B|z−1x|(x)|z−1x|γ−α for x, z ∈ K, it follows that for all y ∈ Bλ(x)  one has  |(R(n,n)(f − fx)(y)| = |〈Πy(f (y)− fx(y)),ϕ(n)  y 〉|  = |〈Πy(f (y)−Γx,y f (x)),ϕ(n)  y 〉|  ≲ ∥Π∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='Bλ(y)∥f ∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='BCr−n (y)  �  α≤α≤γ  r−αn|y−1x|γ−α  ≲ ∥Π∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='Bλ(y)∥f ∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='BCr−n (y)r−αn(|y−1x|  γ−α +r(α−γ)n),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='12)  where C(r) :=  r  r−1 is as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Given n > n0(λ) sufficiently larger, we have a uni-  form bound Cr−n ≤ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' By the convergence of R(n,n) in the first exponent, it follows that R(n)  satisfies the same bound and so inspecting the proof of the second half of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='11, in par-  ticular noticing that we integrate the above estimate over y ∈ Bλ(x), we conclude that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='11)  holds for the limit R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Using the obvious notation we can also rewrite (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5) as  sup  ψ∈Bm  |〈R(f − fx)− ¯R( ¯f − ¯fx),ψλ  x〉| ≲ λγ �  ∥f ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' ¯f ∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='B2λ(x)∥ ¯Π∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='B2λ(x) +∥f ∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='B2λ(x)∥Π− ¯Π∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='B2λ(x)  �  ,  uniformly over λ ∈ (0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' This is seen very similarly to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='11) but using this time that for n  large enough,  |R(n,n)(f − fx)− ¯R(n,n)( ¯f − ¯fx)(y)|  = |〈Πy(f (y)−Γx,y f (x))− ¯Πy( ¯f (y)− ¯Γy,x ¯f (x)),ϕ(n)  y 〉|  = |〈Πy(f (y)−Γx,y f (x)− ¯f (y)+ ¯Γy,x ¯f (x))+(Πy − ¯Πy)( ¯f (y)− ¯Γy,x ¯f (x)),ϕ(n)  y 〉|  ≲  �  ∥Π∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='Bλ(y)∥f ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' ¯f ∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='Bλ(y) +∥Π− ¯Π∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='Bλ(y)∥ ¯f ∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='Bλ(y)  �  �  α≤α≤γ  r−αn|y−1x|  γ−α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  It remains to show that the reconstruction map is unique for γ > 0 and that Rf is indeed an  element of C α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' This is done exactly as in [Hai14, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  30  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2 FUNCTIONS AS MODELLED DISTRIBUTIONS  Let ¯T = (¯T, ¯G) be the polynomial regularity structure equipped with the polynomial model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  We show that in this setting the reconstruction theorem and its inverse map Taylor polyno-  mials to Hölder functions and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For γ > 0 the reconstruction operator is an isomorphism between Dγ(G) and  Cγ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In particular the inverse of the reconstruction map is given by  Cγ(G) ∋ f (·) �→PPPγ  (·)[f ] ∈ Dγ(G),  where PPPa is defined in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Given a modelled distribution in Dγ and since we are working with the polynomial  regularity structure equipped with its canonical model, it follows directly from the bound  satisfied by the image of the reconstruction operator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Equation(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4), and the definition of  Cγ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='13) that the reconstruction operator is a map Dγ(G) → Cγ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Continuity also follows  from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4) and by linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  To see the other direction recall that given a Hölder continuous distribution f ∈ Cγ(G) by  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='22 the {X I f }d(I)<γ are actually functions and, in particular, that ˜Px = ΠxPPP[γ]  x [f ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Therefore, for λ = |x−1y|G  ���Πx  �  PPP[γ]  x [f ]−Γx,yPPP[γ]  y [f ]  �  (ψλ  x)  ��� = |〈 ˜Px − f ,ψλ  x〉|+|〈f − ˜Py,ψλ  x〉| ≲ λγ  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='13)  uniformly in ψ ∈ Br .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' On the other hand, writing PPP[γ]  x [f ]−Γx,yPPP[γ]  y [f ] = �  d(I)<γ cI  x,yηηηI, we find  that  Πx  �  PPP[γ]  x [f ]−Γx,yPPP[γ]  y [f ]  �  (ψλ  x) =  �  d(I)<γ  cI  x,yΠxηηηI(ψλ  x) =  �  d(I)<γ  cI  x,yΠeηηηI(ψλ) =  �  d(I)<γ  cI  x,yλd(I)ΠeηηηI(ψ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Using that Pγ is a finite dimensional vector space and the surjectivity of the linear map  C ∞  c (B1(e)) → RdimPγ,  ψ �→ {ΠeηηηI (ψ)}d(I)<γ  we find that  �  d(I)<γ  |cI  x,y|λd(I) ≲γ sup  ψ∈Br  �����  �  d(I)<γ  cI  x,yλd(I)ΠeηηηI(ψ)  ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='14)  Together, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='13) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='14) imply that that |cI  x,y| ≲ λγ−d(I), we may then apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6 to  see that PPP[γ]  (·)[f ] ∈ Dγ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3 LOCAL RECONSTRUCTION  We will require the following further refinement of the reconstruction theorem which is an  analogue in our case of [Hai14, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  31  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In the setting of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='7 one has the improved bound,  sup  ψ∈Bm  |(Rf −Πx f (x))(ψλ  x)| ≲ λγ∥Π∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='B2λ(x)  sup  y,z∈supp(ψλ  x)  sup  ℓ<γ  |f (z)−Γzy f (y)|ℓ  |y−1z|γ−ℓ  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='15)  as well as, given a second model ¯M( ¯Π, ¯Γ) and a modelled distribution ¯f ∈ Dγ  ¯M, the analogous  bound, that for every λ ∈ (0,1],  sup  ψ∈Bm  |〈RM f −R ¯M ¯f −Πx f (x)+ ¯Πx ¯f (x),ψλ  x〉| ≲ λγ�  ∥f ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' ¯f ∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='supp(ψλ  x)∥ ¯Π∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='B2λ(x)  +∥f ∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='supp(ψλ  x)∥Π− ¯Π∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='B2λ(x)  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='16)  for any x ∈ G and any λ ∈ (0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Since the right hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='15) is linear in f , as in [Hai14] we may assume it to be  equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We use the functions ρ(n) and ϕ(n) from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 and recall that they satisfy  ϕ(n−1)(x) =  �  Br−n+1  ρ(n−1)(y)ϕ(n)  y (x)d y,  ϕ(n−1)  y  =  �  Br−n+1  ρ(n−1)(w)ϕ(n)  yw(x)dw  in particular since  �  ρ(n)(x)dx = 1 we have  �  ϕ(n)  y (·)dy = 1 for all n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Define, for fixed ψλ  x,  the sets  Λn =  �  y ∈ G : supp(ϕ(n)  y )∩supp(ψλ  x) ̸= �  �  ⊂ G  which by definition is contained Bλ+Cr−n(x) with C =  r  r−1 as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='10, as well as a (mea-  surable) function  πn : Λn → supp(ψλ  x)  such that πn(y) ∈ supp(ϕ(n)  y )∩supp(ψλ  x) for every y ∈ Λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Next, let  Rn =  �  Λn  �Rf −Ππn(y) f (πn(y))  �  (ψλ  xϕn  y )d y  = 〈Rf ,ψλ  x〉−  �  Λn  Ππn(y) f (πn(y))(ψλ  xϕn  y )d y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  It follows that for n0 = min{n ∈ N : r−n ≤ λ}, one has  ���(Rf −Πx f (x))(ψλ  x)−Rn  ��� =  ����  �  Λn  �  Πx f (x)−Ππn(y) f (πn(y))  �  (ψλ  xϕn  y )d y  ���� ≲ λγ  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='17)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='as well as for n > n0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='Rn−1 −Rn =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='Λn−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='Ππn−1(y) f (πn−1(y))(ψλ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='xϕn−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='y  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=')d y −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='Λn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='Ππn(z) f (πn(z))(ψλ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='x ϕn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='z )dz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='Λn−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='Ππn−1(y) f (πn−1(y))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='ψλ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='Br−n+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='ρ(n−1)(w)ϕ(n)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='ywdw  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='d y  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='Λn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='Ππn(z) f (πn(z))(ψλ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='xϕn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='z )dz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='Br−n+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='ρ(n−1)(w)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='Λn−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='Ππn−1(y) f (πn−1(y))(ψλ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='xϕ(n)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='yw)d ydw  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='Λn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='Ππn(z) f (πn(z))(ψλ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='xϕn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='z )dz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='Br−n+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='ρ(n−1)(w)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='Λn−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='Ππn−1(zw−1) f (πn−1(zw−1))(ψλ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='xϕ(n)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='z )dzdw  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='Λn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='Ππn(z) f (πn(z))(ψλ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='xϕn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='z )dz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='Br−n+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='ρ(n−1)(w)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='Λn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='Ππn−1(zw−1) f (πn−1(zw−1))−Ππn(z) f (πn(z))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='(ψλ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='xϕn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='z )dzdw .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Since |πn(z)−1πn−1(zw−1)| ≤ ¯Cr−n and for τ ∈ Tα such that |τ| ≤ 1  |Ππn(z)τ(ψλ  xϕn  z )| ≲ λ|s|r−αn ,  we find that  �  Λn  ���  �  Ππn−1(zw−1) f (πn−1(zw−1))−Ππn(z) f (πn(z))  �  (ψλ  xϕn  z )  ���dz ≲ r−γn  and thus conclude  |Rn −Rn−1| ≲ r−γn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  A similar argument gives |Rn| → 0 as n → ∞ which combined with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='17) concludes the proof  of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The proof of the analogous bound, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='16), follows in a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3 SINGULAR MODELLED DISTRIBUTIONS  As in [Hai14] we will eventually be concerned with solutions to SPDEs that take values in  spaces of modelled distributions with permissible singularities in some regions of the do-  main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Our main example will be modelled distributions on space-time domains that are al-  lowed to be discontinuous at {t = 0}, see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' However, as in [Hai14] we build the notion  of singular modelled distributions allowing for singularities on more general sets, generalisa-  tions of which have been used in [GH19a, GH21] to study singular equations with boundary  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  We fix a homogeneous sub-Lie group P ⊂ G with associated Lie algebra for which we write  p ⊂ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The assumption that P be a homogeneous sub-Lie group means that the the scaling  map s restricts to a map ¯s := s|p : p → p which is diagonalisable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We fix a decomposition  33  g = p⊕pc such that pc is also invariant under s and define the homogeneous dimension of P  and its complement, Pc := exp(pc) as,  |¯s| := trace(s|p)  and  |m| = trace(s|pc ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='18)  Furthermore, we set  |x|P := 1∧dG(x,P) = 1∧inf  �  z ∈ P : |x−1z|  �  ,  |x, y|P := |x|P ∧|y|P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Note that since P is closed (being the image under exp of a linear subspace of g), one sees  easily that the infimum above is actually a minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Given K ⊂ G we define the set  KP :=  �  (x, y) ∈ (K\\P)2 : x ̸= y and |x−1y| ≤ |x, y|P  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  That is KP contains all the points in K that are closer to each other than they are to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5 (Singular Modelled Distributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Given a regularity structure T and a sub-  group P as above, for any γ > 0, η ∈ R and maps f : G\\P → T , we set  ∥f ∥γ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K := sup  x∈K\\P  sup  ζ<γ  |f (x)|ζ  |x|(η−ζ)∧0  P  ,  �f �γ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K := sup  x∈K\\P  sup  ζ<γ  |f (x)|ζ  |x|η−ζ  P  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Then given a model M = (Π,Γ) as well as a sector V , the space Dγ,η  P,M(V ) consists of all functions  f : G\\P → V≤γ such that for every compact set K ⊂ G,  ������f  ������  γ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K := ∥f ∥γ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K +  sup  (x,y)∈KP  sup  ζ<γ  |f (x)−Γx,y f (y)|ζ  |y−1x|γ−ζ|x, y|η−γ  P  < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  For two models M = (Π,Γ), ¯M = ( ¯Π), ¯Γ) and two modelled distributions f ∈ Dγ,η  P,M, ¯f ∈ Dγ,η  P, ¯M we  also set,  ������f ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' ¯f  ������  γ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K := ∥f − ¯f ∥γ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K +  sup  (x,y)∈KP  sup  ζ<γ  |f (x)− ¯f (x)−Γx,y f (y)+ ¯Γxy ¯f (y)|ζ  |y−1x|γ−ζ|x, y|η−γ  P  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  If V is a sector of regularity α ∈ A, where appropriate we will use the shorthand Dγ,η  α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='P,M =  Dγ,η  P,M(V ) and we will drop the dependence on the model when the context is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We refer to [Hai14] for more intuition regarding the definition of these spaces  and their properties - all of which carry over to our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In particular [Hai14, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4]  discusses the relationship between the spaces Dγ,η  P  and Dγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The family of norms �f �γ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K and the two following lemmas play a role when we  consider fixed-point maps in Section 4 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Their utility is in allowing us to extract small  scaling parameters in terms of the distance to the subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In the semi-linear evolution  equation setting this allows us to obtain fixed points on sufficiently short time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  34  Lemma3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Let K ⊂ G be a compact domain such that for every x ∈ K and ¯x := argminy∈P |x−1y|  one has that the points ¯x  �  λ·( ¯x−1x)  �  ∈ K for every λ ∈ [0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Also let f ∈ Dγ,η  P  for some γ > 0 and  assume that for every ζ < η the map x �→ Qζ f (x) extends continuously in such a way that for  x ∈ P one has Qζ f (x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Then one has the bound,  �f �γ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K ≲  ������f  ������  γ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K,  where the implied constant depends affinely on ∥Γ∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K but is otherwise independent of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Sim-  ilarly, if ¯f ∈ Dγ,η  P, ¯M with respect to a different model ¯M = ( ¯Π, ¯Γ) and is such that  lim  x→P Qζ(f (x)− ¯f (x)) = 0  for every ζ < η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Then one has the bound,  �f − ¯f �γ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K ≲  ������f ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' ¯f  ������  γ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K +∥Γ− ¯Γ∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K  �������f  ������  γ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K +  ������ ¯f  ������  γ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K  �  ,  with proportionality constant also depending affinely on ∥Γ∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K and ∥¯Γ∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The proof follows almost exactly as that of [Hai14, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For completeness we  provide a sketched proof of the first inequality, the second follows analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Firstly, note that for x ∈ G such that dG(x,P) ≥ 1 or for ζ ≥ η both bounds follow directly  from the definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Hence we restrict our attention to x ∈ K such that dG(x,P) < 1 and  ζ < η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We define a recursive sequence by setting x0 := x, x∞ := ¯x = argminz∈P |x−1z| and  xn := x∞  �  2−n ·(x−1  ∞ x0)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' One has |x−1  n x∞| = |2−n ·(x−1  ∞ x0)| = 2−n|x−1  ∞ x0| as well as  |x−1  n xn+1| ≲G |2−n ·(x−1  ∞ x0)|+|2−(n+1) ·(x−1  ∞ x0)| = 2−n  �3  2  �  |x−1  ∞ x0| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  So using that we can write 2−n|x−1  ∞ x0| = |x−1  n x∞| we find  |x−1  n xn+1| ≲G |x−1  n x∞| = 2−n|x−1  ∞ x0| = 2−ndG(x,P) = 2−n|x|P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='19)  The main difference in the proof is to apply (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='19) in place of [Hai14, Equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4)], then the  rest of the proof adapts closely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' One uses (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='19) together with the definition of  ������f  ������  γ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K, to  show that for any ζ ∈ A,  |f (xn+1)−Γxn+1xn f (xn)|ζ ≲G  ������f  ������  γ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K2n(η−ζ)|x|η−ζ  P  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  To see this, note that for m ≥ η it trivially holds and so for |f (x)|m ≲K |x|η−m  P  , we proceed by  reverse induction, assuming that the required bound holds for all m > ζ and then show that  it also holds for ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Applying the triangle inequality, the inductive hypothesis, the properties  of Γ and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='19) we find, as in the proof of [Hai14, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5], that  |f (xn+1)− f (xn)|ζ ≲ 2n(η−ζ)|x|η−ζ  P  ,  where the constant depends affinely on ∥Γ∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Then, using the assumption that Qζ f (x) = 0  for all ζ < η and x ∈ P and applying the above inequality we find,  |f (x)|ζ = |f (x)− f (x∞)|ζ ≤  �  n>0  |f (xn+1)− f (xn)|ζ ≲K |x|η−ζ  P  �  n≥0  δn(η−ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Using that A is locally finite we may complete the induction which finishes the proof of the  first inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The proof of the second follows as in [Hai14, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  35  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Let γ > 0, κ ∈ (0,1) and assume f , ¯f satisfy the assumptions of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Then, for every compact K ⊂ G, one has  ������f ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' ¯f  ������  (1−κ)γ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K ≲ �f − ¯f �κ  γ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K  �������f  ������  γ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K +  ������ ¯f  ������  γ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K  �1−κ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Follows by a direct adaptation of the proof of [Hai14, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' One need only replace Rd  there by G here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 RECONSTRUCTION THEOREM FOR SINGULAR MODELLED DISTRIBUTIONS  Since the reconstruction is purely local, it follows from our earlier proof that for any singular  modelled distribution, f ∈ Dγ,η  P , there exists a unique element ˜Rf ∈ S′(G\\P), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' in the dual  of smooth functions compactly supported away from P, such that,  〈 ˜Rf −Πx f (x),ψλ  x〉 ≲ λγ,  for all x ∉ P and λ ≪ dG(x,P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' However, we show below that under appropriate assumptions  there exists a natural extension of ˜Rf to an actual distribution on G with regularity Cα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='18 (Singular Reconstruction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Let f ∈ Dγ,η  P (V ), γ > 0 and η ≤ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Then, provided  α ∧ η > −m, where m is the scaled dimension of the complement of the singular hyperplane  defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='18), there exists a unique distribution Rf ∈ Cα∧η  s  such that (Rf )(ϕ) = ( ˜Rf )(ϕ)  for any smooth test function compactly supported away from P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' If f and ¯f are given with  respect to two models M and ¯M then, for any compact K, it holds that  ∥RM f −R ¯M ¯f ∥Cα∧η(K) ≲  ������f ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' ¯f  ������  γ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' ¯K +  ������M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' ¯M  ������  γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' ¯K,  where the constant depends on semi-norms of f , ¯f and Z, ¯Z on ¯K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  We provide the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='18 at the end of this section, let us first make some  preparatory observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Recalling the decomposition g = p ⊕pc defined at the start of the  subsection, we define the projections πc : g → pc and πp : g → p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Then using the decomposi-  tion X = X p + X c ∈ p⊕pc, we define the map  ˜Φ : g → G,  ˜Φ(X ) = exp(X p)exp(X c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='20)  Similarly to Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='18 one sees that ˜Φ is a global diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We then define the map  NP : G → R+,  x �→  ��exp  �  πc ◦ ˜Φ−1(x)  ���  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='21)  and observe the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  For x ∈ G one has that NP(x) = 0 if and only if x ∈ P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' This follows from the fact that  P = exp(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  For x ∈ G and δ > 0 one has the identity  NP(δ· x) = δNP(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='22)  Indeed, writing x = ˜Φ( ˜X ), we have  NP(δ· x) = |exp(πc(Dδ ˜X ))| = |exp(Dδ πc( ˜X ))| = |δ·exp(πc( ˜X ))| = δNP(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  36  For any x ∈ G and y ∈ P one has  NP(yx) = NP(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='23)  This follows from the observation that, writing x = ˜Φ( ˜X ) and y = ˜Φ(Y ) = exp(Y ) one  finds that yx = exp(H(Y , ˜X p))exp( ˜X c) where H was defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Identity (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='23) then  follows from the fact that H(Y , ˜X p) ∈ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  The map NP is Lipschitz continuous on G and it follows from Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4 that NP is  smooth on G\\P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  There exists a constant C > 0 such that for all x ∈ G  CNP(x) ≤ dG(x,P) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='24)  In a neighbourhood of the origin this follows directly from the fact that NP is Lipschitz  continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Homogeneity of Np (given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='22) above) and of the distance function  x �→ dG(x,P) then shows that this constant is in fact uniform on all of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  For all x ∈ G  dG(x,P) ≤ NP(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='25)  Indeed, write x = ˜Φ(X ) = exp(X p)exp(X c) and note that  d(x,P) ≤ |exp(X p)−1x| = |exp(X c)| = NP(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The proof is analogous to that of [Hai14, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='9], the only ele-  ment that does not adapt ad verbatim, is the construction of the partition of unity ϕx,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We  therefore present an alternative construction of a partition of unity on G, which satisfies all  the required conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  First let ϕ : R+ → [0,1] be a smooth compactly, supported function such that supp(ϕ) =  [1/2,2] and with the property that for all r ∈ R+,  �  n∈Z  ϕ(2nr) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Secondly, let Z ⊂ P be a lattice (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3) and let ˜ϕ be smooth, compactly supported,  such that  �  y∈Z  ˜ϕy(x) = 1  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='26)  for all x in the 2  C -fattening of P ⊂ G, where C is the constant in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='24)and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For y ∈ Z we  then set  φy(x) := ϕ(CNP(x)) ˜ϕy(x) ,  where the constant C is as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' To conclude, we then set for every n ≥ 0, y ∈ Z and x ∈ G,  φn,y(x) := φy(2n · x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  37  By using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='22) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='23), it directly follows that φn,y(x) = (φ1,e)  �  y−1(2n · x)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Since φ1,e has  compact support and is such that for all x ∈ supp(φ1,e), one has dG(x,P) ≥ CNP(x) ≥ 1/2, it  only remains to check that {φn,y}n∈Z,y∈Z is in fact a partition of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Indeed let x ∈ G, then  �  n∈Z,y∈Z  φn,y(x) =  �  n∈Z,y∈Z  ϕ(CNP(2n · x)) ˜ϕy(2n · x) =  �  n∈Z  ϕ(2nCNP(x))  �  y∈Z  ˜ϕy(2n · x)  �  ��  �  =1 given d(x,P)≤ 1  C 2−n+1  = 1 ,  where we used (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='26) in the last equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The remainder of the proof of [Hai14, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='9] then  adapts ad verbatim by also also making use of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' If the model M is smooth, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Πxτ ∈ C ∞(G) for every τ ∈ T, one finds exactly as  in [Hai14, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='15] that for any modelled distribution f ∈ Dγ with γ > 0 one has the identity  RM f (x) =  �  Πx f (x)  �  (x) (and in particular RM f is a continuous function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4 LOCAL OPERATIONS  To handle SPDEs using regularity structures we require suitable extensions to modelled dis-  tributions of standard local operations such as differentiation, multiplication and composi-  tion with smooth functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' These extensions adapt easily from [Hai14], thus for brevity we  provide them directly for singular modelled distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 DIFFERENTIATION OF SINGULAR MODELLED DISTRIBUTIONS  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Given a sector V of a regularity structure T = (T,G), a linear operator ∂ : V → T  defines an abstract differential operator of homogeneous degree β, if  for any a ∈ Vα it holds ∂a ∈ Tα−β,  for any a ∈ V and Γ ∈ G, one has Γ∂a = ∂Γa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  We say ∂ realizes L for the model M = (Π,Γ) if  for any a ∈ V and x ∈ G, one has Πx∂a = LΠxa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  The following proposition is an immediate consequence of the definitions and the unique-  ness of the singular reconstruction operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In the setting of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6 let f ∈ Dγ,η  P (V ) for some γ > η, then ∂f ∈  Dγ−β,η−β  P  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Furthermore, if the sector V has regularity α and it holds that γ > β as well as  α∧η > β−m, then one has R∂f = LRf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For the first claim one may directly adapt the proofs of [Hai14, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='28 & Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  The second claim follows by applying the Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='18 to Dγ−β,η−β  P  ,  38  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We note that any left invariant differential operator L of homogeneous degree  β on G is of the form  L =  �  d(I)=β  aI X I,  for some I ∈ Nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Thus, in order to lift such an L, it suffices to lift each of the differential  operators {Xi }d  i=1 to abstract differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Recall that in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 we lifted the differential operators Xi crucially using  the additional structure of the polynomial regularity structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' This is by no means the only  lift, though certainly the most canonical one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' One observes that it is always possible to extend  a regularity structure and a model such that it carries some lift of a given differential operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2 MULTIPLICATION AND COMPOSITION WITH SMOOTH FUNCTIONS  We recall the notion of a product on a regularity structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Given a regularity structure T = (T,G) and two sectors V,W ⊂ T, a continuous  bilinear map  ⋆ : (V,W ) → T,  (a,b) �→ a ⋆b  defines a product if  one has a ⋆b ∈ Tα+β for every α, β ∈ A and a ∈ Tα, b ∈ Tβ,  Γ(a ⋆b) = (Γa)⋆(Γb) for every Γ ∈ G and for every a ∈ V, b ∈ W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  We say that a sector V is stable under the product if V ⋆V ⊆ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Given a product ⋆ and any  γ ∈ R, we introduce the truncated product ⋆γ, which is given by the composition of ⋆ with the  projections onto T<γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  We show that the product of two singular modelled distributions is again a singular mod-  elled distribution with possibly new regularity and singularity parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Let P be a hyperplane as described above and f1 ∈ Dγ1,η1  P  (V1) and f2 ∈  Dγ2,η2  P  (V2) for two sectors V1, V2 of respective regularities α1, α2 ∈ A and let ⋆ be a product  on (V1, V2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Then f = f1 ⋆γ f2 ∈ Dγ,η  α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='P with  α = α1 +α2,  γ = (γ1 +α2)∧(γ2 +α1),  η = (η1 +α1)∧(η2 +α1)∧(η1 +η2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Here ⋆γ is the projection of ⋆ onto T<γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Furthermore, for products between modelled distributions as above but on differing models,  writing f = f1 ⋆γ f2 and g = g1 ⋆γ g2 we have the bound,  ������f ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='g  ������  γ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K ≲  ������f1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='g1  ������  γ1,η1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K +  ������f2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='g2  ������  γ2,η2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K +∥Γ− ¯Γ∥γ1+γ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K,  uniformly over any compact set K ⊂ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' One may directly adapt the proofs of [Hai14, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='7] and [Hai14, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='12], recalling  that the homogeneous distance satisfies a triangle inequality with a constant, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Remark2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  39  We define the composition of modelled distributions with smooth functions as in [Hai14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Given a ∈ V , a function-like sector, we decompose a = ¯a1 + ˜a with ˜a ∈ T>0 and ¯a = 〈1,a〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Further, let ζ > 0 be the smallest non-zero value such that Vζ ̸= 0 so that we actually have  ˜a ∈ T≥ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Given a smooth function F : Rn → R, for some n ≥ 1, and a function-like sector V which is  stable under some product ⋆γ, we lift F to a function ˆFγ : V n → V by the formula,  ˆFγ(a) :=  �  k  DkF(Q0a)  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  ˜a⋆γk,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='27)  where we used the isomorphism T0 ∼ R for each component of a = (a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=',an) ∈ V n, as well as  the notation ˜ai = ai −Q0ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Here the sum runs over all possible multi-indices and we extend  the product ⋆ in a natural way for vectorial arguments and multi-index powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  We make the same observation as in [Hai14, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2], that although the sum in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='27) looks  infinite, since ζ > 0 we have ˜a⋆γk ∈ T≥|k|ζ and so only finitely many terms of the sum are  non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In the following proposition we naturally extend the definition of a modelled dis-  tribution and associated norms componentwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Let P be as above, γ > 0 and f = (f1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=', fn) ∈ Dγ,η  P (V n) be a collection of  modelled distributions for some function-like sector V which is stable under the product ⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Let  furthermore F ∈ C∞(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='R), then, provided η ∈ [0,γ], the modelled distribution  ˆFγ(f )(x) : G → V,  with ˆF(f ) defined as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='27), belongs to Dγ,η  P (V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Furthermore, the map ˆFγ : Dγ,η  P (V ) →  Dγ,η  P (V ) is locally Lipschitz continuous in any of the semi-norms ∥ · ∥γ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K and |||·|||γ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Furthermore, the analogous Lipschitz bound holds when working with two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' One may follow ad verbatim the proof of [Hai14, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='13], as well as [HP15, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='11]  for the last sentence, since all Taylor expansions are carried out in Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5 CONVOLUTION WITH SINGULAR KERNELS  In this section we describe how to lift the action of singular kernels onto the regularity struc-  ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' While most arguments adapt from [Hai14] some care has to be taken due to the fact  that convolutions are not commutative and we do not have an explicit formula for Taylor ex-  pansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' This latter issue is circumvented using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6 which allows us to reduce our  analysis to similar expressions as appear in [Hai14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The examples we have in mind are the  singular part of Greens functions of left invariant differential operators satisfying the follow-  ing assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The kernel K : G\\{e} → R can be decomposed as  K (x) =  �  n∈N  Kn(x)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='28)  where the smooth functions Kn are supported on B2−n and  40  for each I ∈ Nd there exists a constant C(I) > 0, uniform in n ∈ N such that  sup  x∈G  |X IKn(x)| ≤ C(I)2(|s|−β+d(I))n,  for any multi-indices I, J ∈ Nd there exists a constant C(I, J) > 0 uniform in n ∈ N such  that,  ����  �  G  ηI(x)X JK (x)dx  ���� ≤ C(I, J)2−βn,  there exists an integer r, such that  �  G  ηI(x)K (x)dx = 0  for all multi-indices I ∈ Nd with scaled degree d(I) ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We note that all of the analysis in the remainder of this section also applies  to kernels of the form K : (G \\{e})2 → R satisfying an analogue of [Hai14, Ass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 & Ass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4]  adapted to our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Although Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='25 is somewhat less general we choose to  work with it for two reasons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' firstly it is simpler to verify and secondly it highlights the role that  translation invariance plays in our applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4 which is an amalgam of [Hai14,  Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5 & Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='7] in our setting, shows that fundamental solutions of left-translation in-  variant, homogeneous linear operators can always be decomposed into a compactly support  part satisfying Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='25 and a sufficiently well-behaved remainder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Although we work explicitly with the one-parameter kernels of Assumption  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='25 it will sometimes convenient in the proofs below to define K (x, y) := K (y−1x) and use  the notation K (f )(x) :=  �  K (x, y)f (y)dy = f ∗K (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Note that under this convention, for any  left-translation invariant vector field X , one has (X K )(f ) = X (K f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  From now on we shall exclusively work with regularity structures T models M satisfying  the following assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For each a ∈ △, the vector space Ta coincides with the linear span of abstract  monomials ηηηI with d(I) = a and the model M ∈ MT restricted to the polynomial sector ¯T =  �  a∈△Ta, is the canonical polynomial model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  We point out that the assumption K annihilates polynomials causes no real restriction on  the type of kernels since the result [Hai14, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5] adapts in a straightforward manner to  our setting, see also Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We point out that this assumption is convenient but  not crucial for the theory, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' [HSa].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In the remainder of this subsection we show how the  action of kernels of this type are lifted onto the regularity structure and act on modelled dis-  tributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Given a γ ∈ R∩△ we write Kγ for this lift;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' it corresponds to K in the sense that for  f ∈ Dγ  RKγ f = K (Rf )  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='29)  and in that it satisfies an appropriate version of the classical Schauder estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  41  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Given a sector V , a map I : V �→ ¯T is called an abstract integration map of  order β > 0 if it satisfies the following properties:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For each α ∈ A, I : Vα → Tα+β, where Tα+β := {0} for α+β ∉ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' It annihilates polynomials, that is I : ¯T∩V → {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For each τ ∈ T, Γ ∈ G : (I Γ−ΓI)(τ) ∈ ¯T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Assume that the kernel K satisfies Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='25 for some β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We associate to K the  map J : Rd → L(T, ¯T) which for every τ ∈ Tα is given by  J (x)τ :=  �  n  PPPα+β  x  (Πxτ∗Kn),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='30)  where the last sum is seen to converge absolutely by first observing that by Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='25  for any τ ∈ Tα  |X I(Πxτ∗Kn)(x)| = |Πxτ∗ X IKn(x)| = |Πxτ(X I  1Kn(x,·))| ≲ 2−n(α+β−d(I)) ,  and then using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Given a regularity structureT equipped with an Integration map I, a kernel K  and a model M = (Π,Γ), we say the model M realises K for I if for each τ ∈ Tα and each x ∈ G,  ΠxIτ = K (Πxτ)−ΠxJ (x)τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Now we can define the lift of the kernel K, namely for f ∈ Dγ(V ) we set:  Kγ f (x) := I f (x)+J (x)f (x)+Nγ f (x),  where  (Nγf )(x) =  �  n  PPPγ+β  x  �  (Rf −Πx f (x))∗Kn  �  where the last sum converges by the same argument as for J (x)τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Given a kernel K satisfying Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='25 a regularity structure and model  satisfying Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='28, it turns out that one can always extend the regularity structure  and model to be equipped with an integration map I realizing the kernel K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' This is the con-  tent of the extension theorem found as [Hai14, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='14], which holds in our setting as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  While we do not reproduce the whole proof since it is a straightforward adaptation of the  original one, we present the main steps below, see Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='31, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='32 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='33, so that the  interested reader will easily be able to fill in the remaining details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  The next theorem which is an analogue of [Hai14, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='12], confirms that Kγ does indeed  correspond to K in the sense of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='29) and satisfies the desired Schauder estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  42  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Let γ ∈ R \\△ and β > 0 be such that γ + β ̸∈ △, let K : G \\{e} → R be a kernel  satisfying Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='25 for r ≥ γ+β, let T = (T,G) be a regularity structure and M = (Π,Γ)  be a model satisfying Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Furthermore assume that T is equipped with an ab-  stract integration operator and M realises K for I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Then for any sector V of regularity α ∈ A the  operator Kγ is a continuous linear map from Dγ  M(V ) to Dγ+β  (α+β)∧0 satisfying the identity  RKγ f = K (Rf ),  for all f ∈ Dγ  M(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Furthermore, if we denote by M = (Π,Γ), ¯M = ( ¯Π, ¯Γ) two admissible models  and by Kγ, respectively ¯Kγ the associated operators, one has the bound  ∥Kγ f ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' ¯Kγ ¯f ∥γ+β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K ≲C  ������f , ¯f  ������  γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' ¯K +∥Π− ¯Π∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' ¯K +∥Γ− ¯Γ∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' ¯K ,  where the implicit constant depends on the norms of M, ¯M and f ∈ Dγ  M(V ) and ¯f ∈ Dγ  ¯M(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  The fact that the next lemma ([Hai14, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='16]) still holds in our setting is the underlying  reason that all proofs extend in a rather straight forward manner from [Hai14] and one does  not require a more involved notion of abstract integration map, which is for example the case  on general Riemannian manifolds c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' [HSa].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In the setting of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='30 one has the identity  Γx,y(I +J (y)) = (I +J (x))Γx,y  for all x, y ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The proof consists of unravelling the Definitions and using the fact that the map Πx is  injective when restricted to polynomials, exactly as in [Hai14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  We introduce the following quantity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' for I ∈ Nn,α > 0,n ∈ N and x, y,z ∈ G, set  K I,α  n,xy(z) := X I  1Kn(y,z)− ˜Pα+β−d(I)  x  [X I  1Kn(·,z)](y) = X I  1  �  Kn(·,z)− ˜Pα+β  x  [Kn(·,z)]  �  (y)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='31)  where the second equality follows from Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Here we reiterate that we are using the  notation K (x, y) := K (y−1x) where K satisfies Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Taylor’s theorem and Remark  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='15 then yield that  K I,α  n,xy(z) =  �  |J|≤[a]+1,d(J)≥α+β−d(I)  �  G  X J  1(X I  1Kn)(x ˜z,z)Q J(x−1y,d ˜z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='32)  43  As in [Hai14] the motivation for defining these quantities comes from the identity  Πx(Iτ)(φ) = K (Πxτ)(φ)−Πx J(x)τ(φ)  =  �  n  ��  Πxτ(Kn(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='·))− ˜Pα+β  x  [Kn(Πxτ)](y)  �  φ(y)d y  =  �  n  ��  Πxτ(Kn(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='·))− ˜Pα+β  x  �  (Πxτ)2(Kn(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='·))  �  (y)  �  φ(y)d y  =  �  n  ��  Πxτ(Kn(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='·))−(Πxτ)(˜Pα+β  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 [Kn(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='·)](y)  �  φ(y)d y  =  �  n  �  Πxτ  �  Kn(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='·)− ˜Pα+β  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 [Kn(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='·)](y)  �  φ(y)d y  =  �  n  �  Πxτ(K 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='α  n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='xy)φ(y)d y  where we use the subscript in (Πxτ)2(Kn(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='·)) to clarify that this denotes the function w �→  Πxτ(Kn(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='·)) and the analogous subscript for ˜Px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1[K (·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='·)](y) to clarify that one expands in the  first coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The next Lemma collects the results of [Hai14, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='18, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='19], the  proofs of which adapt ad verbatim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Let α ∈ A and τ ∈ Tα and assume α+β ∉ △, then the following bound holds  |(Πyτ)(K I,α  n,xy)| ≲ ∥Π∥α,B2(x)(1+∥Γ∥α,B2(x))  �  δ>0  2δn|y−1x|δ+α+β−d(I) ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='33)  where the sum over δ runs over a finite set of positive real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The same bound holds for  |(Πxτ)(K I,α  n,xy)|, as well as the analogue bound on the difference of two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Furthermore one has the bound  �  n  ����  �  G  (Πxτ)(K I,α  n,xy)φλ  x(y)dy  ���� ≲ λα+β∥Π∥α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='B2(x)(1+∥Γ∥α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='B2(x)) ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='34)  as well as the analogous bound for the difference of two models, where all these bounds hold  uniformly over x ∈ G, λ ∈ (0,1] and φ ∈ Br  As in [Hai14] we introduce for x, y ∈ G the following operator  Jx,y := J (x)Γx,y −Γx,yJ (y)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='35)  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Under the assumptions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='30 one has for each a ∈ ∆, τ ∈ Tα  |Jx,yτ|a ≲ ∥Π∥α,B2(x)(1+∥Γ∥α,B2(x))|y−1x|α+β−a|τ|  uniformly in x, y ∈ G satisfying x ∈ B1(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Again, the analogous bound holds for the difference  of two models holds as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' First we write Jx,y = �  n J n  x,y where each J n  x,y is defined by replacing K by only one  summand Kn in the definition of Jx,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Observe that since J n  x,yτ ∈ ¯T<α+β it suffices by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6  to show that  pn  τ (y) := Πe(J n  x,yτ)(y) ∈ Pα+β  44  satisfies  �  n  |(X I pn  τ )(e)| ≲ ∥Π∥α,B2(x)(1+∥Γ∥α,B2(x))|y−1x|α+β−d(I)|τ| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='36)  We treat the cases |y−1x| ≤ 2−n and |y−1x| > 2−n separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In the case |y−1x| ≤ 2−n, using  the definitions of the polynomial regularity structure, we find  pn  τ (x−1z) = Πe(J n  x,yτ)(x−1z) = Πx(J n  x,yτ)(z)  and thus  X I pn  τ (e) = X I(ΠxJ n  x,yτ)(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Using the analogous notation J n(x) for the operator consisting only of one summand in  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='30),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' we find  ΠxJ n(x)Γx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='yτ(z) =  �  γ≤α  Πx  �J n(x)QγΓx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='yτ  �  (z)  =  �  γ≤α  ΠxPγ+β  x  [  �  ΠxQγΓx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='yτ  �  2Kn(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='·)](z)  =  �  γ≤α  ˜Pγ+β  x  [  �  ΠxQγΓx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='yτ  �  2Kn(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='·)](z)  =  �  γ≤α  ΠxQγΓx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='yτ  �  ˜Pγ+β  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 [Kn(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='·)](z)  �  = (Πyτ)  �  ˜Pα+β  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 [Kn(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='·)](z)  �  −  �  γ≤α  ΠxQγΓx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='yτ  �˜Pα+β  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 [Kn(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='·)]− ˜Pγ+β  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 [Kn(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='·)]  �  (z)  = ˜Pα+β  x  [(Πyτ)2Kn(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='·)](z)−  �  γ<α  ΠxQγΓx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='yτ  �˜Pα+β  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 [Kn(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='·)]− ˜Pγ+β  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 [Kn(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='·)]  �  (z)  and  ΠxΓx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='yJ n(y)τ(z) = ΠyJ n(y)τ(z) = ˜Pα+β  y  [(Πyτ)2Kn(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='·)](z)  = ˜Pα+β  x  �  ˜Pα+β  y  [(Πyτ)2Kn(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='·)]  �  (z)  = ˜Pα+β  x  ��  Πyτ  �  2  �˜Pα+β  y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 [Kn(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='·)]  ��  (z) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  where in the third equality we used the fact that ˜Pα+β  x  acts as the identity map on polynomial  functions of degree less then α+β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Therefore,  pn  τ (x−1z) = ˜Pα+β  x  �  (Πyτ)2  �  Kn(·,·)− ˜Pα+β  y,1 [Kn(·,·)]  ��  (z)  �  ��  �  :=pn  τ,1(x−1z)  −  �  γ<α  ΠxQγΓx,yτ  �˜Pα+β  x,1 [Kn(·,·)]− ˜Pγ+β  x,1 [Kn(·,·)]  �  (z)  �  ��  �  :=pn,γ  τ,2 (x−1z)  Using the general formula X I(˜Pa  x[f ])(x) = X I(Pa  x[f ])(e) = X I f (x) for d(I) < a we find that  X I pn  τ,1(e) = X I�  (Πyτ)2  �  Kn(·,·)− ˜Pα+β  y,1 [Kn(·,·)]  ��  (x) = (Πyτ)(K I,α+β  n,x,y )  and so by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='32 we find that  �  n :|y−1x|≤2−n  |X I pn  τ,1(e)| ≲ |y−1x|α+β−d(I)|τ| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  45  Concerning pn,γ  τ,2 , since one has the identity  pn,γ  τ,2 (x−1z) = −˜Pγ+β  x  [(ΠxQγΓx,yτ)2Kn(·,·)](z)+ ˜Pα+β  x  [(ΠxQγΓx,yτ)2Kn(·,·)](z) ,  we find that X I pn,γ  τ,2 (e) = 0 unless d(I) ∈ [γ+β,α+β), in which case  X I pn,γ  τ,2 (e) = ΠxQγΓx,yτ  �  X I  1Kn(x,·)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Thus,  �  n :|y−1x|≤2−n  |X I pn,γ  τ,2 (e)| ≲  �  n :|y−1x|≤2−n  |y−1x|α−γ2−n(γ+β−d(I))|τ|a ≲ |y−1x|α+β−d(I)|τ| ,  where we used that the case d(I) = γ+β does not arise as by Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='28  we have γ+β ∉ △.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' This concludes the case |y−1x| > 2−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  To treat the case |y−1x| > 2−n, one writes  pn  τ (z) =  �  γ≤α  ΠxJ n(x)(QγΓx,yτ)(xz)  �  ��  �  =:qn,γ  τ,1 (z)  −ΠxΓx,yJ n(y)τ(xz)  �  ��  �  =:qn  τ,2(z)  and using the fact that γ+β ∉ △ one finds  |X I qn,γ  τ,1 (e)| =  ���ΠxQγΓx,yτ  �  X I  1Kn(x,·)  ���,  if d(I) < γ+β,  0,  otherwise,  which is bounded uniformly over |y−1x| < 1 by a constant multiple of |y−1x|α−γ2−n(γ+β−d(I)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Concerning qn  τ,2, we find that  qn  τ,2(z) = ΠxΓx,yJ n(y)τ(xz) = ΠyJ n(y)τ(xz) = Pγ+β  y,1 [(Πyτ)2Kn(·,·)](y−1xz))  and therefore by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5  |X I qn  2 (e)| ≲  �  d(I)<d(J)≤α+β  sup  d(J′)≤d(J)  ���Πyτ  �  X J′  1 Kn(y,·)  ���� |y−1x|d(J)−d(I)|τ|  ≲  �  d(I)≤d(J)<α+β  sup  d(J′)≤d(J)  2−n(α+β−d(J′))|y−1x|d(J)−d(I)|τ|  ≲  �  d(I)≤d(J)<α+β  2−n(α+β−d(J))|y−1x|d(J)−d(I)|τ| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Since pn  τ = �  γ≤α qn,γ  τ,1 + qn  τ,2 one obtains (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='36) by summing over  �  n ∈ N : |y−1x| > 2−n�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' First, we need to check that for a ∈ A it holds that  |Kγ f (x)−Γx,yKγ f (y)|a ≲ |y−1x|γ+β−a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='37)  For a ∉ △ this bound follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='33 and properties of modelled distributions by  an ad verbatim adaptation of the same argument in the proof of [Hai14, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The only  46  place, where the proof [Hai14, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='12] does not adapt directly is when showing the bound  for a ∈ △.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' As in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='33 we use Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6 to circumvent the difficulty of  not having explicit Taylor expansions in our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The rest of the proof adapts almost ad  verbatim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  We first note that the polynomial part of Kγ f (x)−Γx,yKγ f (y) is given by  P := Γx,yNγf (y)  �  ��  �  =:P1  −Nγf (x)  � �� �  =:P2  +J (x)(Γx,y f (y)− f (x))  �  ��  �  =:P3  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='38)  and thus, in order to prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='37) for the polynomial part it suffices by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6 to show  that for any d(I) < γ+β, p(e) := ΠeP satisfies  |X I p(e)| ≲ |y−1x|γ+β−d(I) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='39)  Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='28) we define the analogous decompositions J = �  n J n and Nγ = �  n N n  γ and simi-  larly write P = �  n=0 Pn, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  As usual we use different strategies for small and large scales, starting with the case 2−n ≤  |y−1x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In this case we separately estimate pi(e) := ΠePi for i ∈ {1,2,3} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Noting that  pn  1 (z) = ΠxΓx,yNγf (y)(xz) = ΠyNγ f (y)(xz) = Pγ+β  y,1  �  (Rf −Πy f (y))2  �  Kn(·,·)  ��  (y−1xz),  we find by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5  |X I pn  1 (e)| ≲  �  d(I)≤d(J)<γ+β  sup  d(J′)≤d(J)  ���  �Rf −Πy f (y)  ��  X J′  1 Kn(y,·)  ����|y−1x|d(J)−d(I)  ≲  �  d(I)≤d(J)<γ+β  sup  d(J′)≤d(J)  2−n(γ+β−d(J′))|y−1x|d(J)−d(I)  ≲  �  d(I)≤d(J)<γ+β  2−n(γ+β−d(J))|y−1x|d(J)−d(I) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Regarding pn  2 we find that X I pn  2 (e) =  �Rf −Πx f (x)  ��  X I  1Kn(x,·)  �  and thus by the Re-  construction, see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='7,  |X I pn  2 (e)| ≲ 2−n(γ+β−d(I)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Lastly, for pn, using the definition of J (x) and properties of models and modelled dis-  tributions we find  |X I pn  3 (e)| ≤  �  δ∈A: d(I)−β<δ<γ  ���  ΠxQδ(Γx,y f (y)− f (x)  �  (X I  1K (x,·))  ��  ≲  �  δ∈A: d(I)−β<δ<γ  |y−1x|γ−δ2−n(β+δ−d(I)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  47  Therefore summing over 2−n ≤ |y−1x|, since β+δ ∉ △ for any δ ∈ A and β+γ ∉ △, we find  �  n :2−n≤|y−1x|  |X I pn(e)| ≤  �  n : 2−n≤|y−1x|  3�  i=1  |X I pn  i (e)| ≲ |y−1x|γ+β−d(I) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Next we turn to the case 2−n > |y−1x|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' here a computation analogous to the one preceding  [Hai14, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='48)] gives  X I pn(e) = (Πy f (y)−Rf )(K I,γ  n,x,y)−  �  ζ≤d(I)−β  �  ΠxQζ(Γx,y f (y)− f (x)  ��  X I  1Kn(x,·)  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='40)  which can be bounded exactly the way it is done there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Finally, one may follow the concluding steps of the proof of [Hai14, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='12], finding that  for any φ ∈ B,  �  Πx(Kγ f (x))−K (Rf )  �  (φλ  x) =  �  n  ��  Πx f (x)−Rf  �  (K γ  n,xy)φλ  x(y)d y  and thus one obtains the identity RKγ f = K (Rf ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  The proof of the difference bound follows by similar steps as to those presented here and  applied in the proof of [Hai14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6 SCHAUDER ESTIMATES FOR SINGULAR MODELLED DISTRIBUTIONS  Since our main application will be to semi-linear evolution equations we will often require  a Schauder estimate for modelled distributions with permissible singularities near a given  subgroup P ⊂ G, as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3 on singular modelled distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In the setting of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='30, given a sector V of regularity  α ∈ A, the operator Kγ is well defined on Dγ,η  P (V ) for η < γ provided that γ > 0 and η∧α > −|m|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Furthermore, if γ+β ∉ △ and η+β ∉ △, one has Kγ f ∈ Dγ+β,(η∧α)+β  P  continuity bound  ������Kγ f ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' ¯Kγ ¯f  ������  γ+β,(η∧α)+β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='K ≲  ������f ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' ¯f  ������  γ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' ¯K +∥Π− ¯Π∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' ¯K +∥Γ− ¯Γ∥γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' ¯K  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='41)  over any compact set K ⊂ G and admissible models, where the implicit constant is uniform in  the semi-norms of M, ¯M and f ∈ Dγ,η  P,M(V ), ¯f ∈ Dγ,η  P, ¯M(V ) on ¯K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The proof follows exactly along the same lines as the one of [Hai14, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='16], where  as in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='30 the only modification needed is due to the fact that our Taylor  expansions are non explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Using the same notation as in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='30, we  recall (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='38)  P := Γx,yNγf (y)  �  ��  �  =:P1  −Nγ f (x)  � �� �  =:P2  +J (x)(Γx,y f (y)− f (x))  �  ��  �  =:P3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='42)  This time, in order to show that Kγ f ∈ Dγ+β,(η∧α)+β  P  , by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6, we are required to show  that for p(e) := ΠeP = �3  i=1ΠePi,  |X I p(e)| ≲ |y−1x|  γ+β−d(I)|x, y|(η∧α)+β−γ  P  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='43)  There are three scales, which need separate arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  48  In the case 2−n ≤ |y−1x| one proceeds exactly as in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='30 using the  decomposition pn = �3  i=1 pn  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  In the cases 2−n ∈  �  |y−1x|, 1  2|x, y|P  �  and 2−n ≥ 1  2|x, y|P one uses Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='40) and  proceeds exactly as in [Hai14] from there onwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Again, the proof of the difference bound follows analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='7 SYMMETRIES  As in [Hai14, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6], we shall consider modelled distributions which respect certain sym-  metries of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In this work we will only consider the symmetries of G under the canonical left  action of a discrete subgroup G ⊂ G as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3 acting by G×G ∋ (n,x) �→ (nx) ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We  extend this to an action on function ψ : G → R by pull-back, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  (n∗ψ)(x) := ψ(n−1x) = ψn(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  For a regularity structure T = (T,G) we give the following definition, by analogy with [Hai14,  Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='33] but restricted to this more straightforward setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Given a discrete sub-group G ⊂ G as above, we say that a model M = (Π,Γ) is  adapted to the action of G if, for every test function, φ ∈C ∞  c (G), x ∈ G, τ ∈ T and n ∈ G one has  (Πnxτ)(n∗ψ) = (Πxτ)(ψ)  and  Γnx,ny = Γx,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  A modelled distribution f : G → T is said to be symmetric if f (nx) = f (x) for every x ∈ G and  n ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  The following proposition is an amalgam of [Hai14, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='38 & Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Let T be a regularity structure, G ⊂ G be a discrete subgroup as above and  M = (Π,Γ) be adapted to the action of G (according to Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Then, for every mod-  elled distribution f ∈ Dγ, for some γ > 0, symmetric with respect to the action of G, the follow-  ing hold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For every φ ∈C ∞  c (G) and n ∈ G one has (Rf )(n∗φ) = (Rf )(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For any x ∈ G and n ∈ G one has (Kγ f )(nx) = (Kγ f )(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The proof of the first point follows exactly the steps of the proof of [Hai14, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='38]  in our simplified setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  The proof of the second follows similarly along the lines of the proof of [Hai14, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='23],  in particular using the assumption that the model is adapted to the action of G and the vector  fields {Xi}d  i=1 are left-translation invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  49  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='8 BOUNDS ON MODELS  We state two results which, as in the Euclidean setting, allow one to reduce the number of  stochastic estimates needed in order to obtain convergence of models to only Π|T<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The  proof of the next proposition, [Hai14, Prop 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='31], adapts ad verbatim to our setting  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Let T = (T,G) be a regularitystructure and (Π,Γ) a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For α ∈ (0,∞)∩A,  the action of Π on Tα is fully determined by the action of Π on T<α as well as the action of Γ on  Tα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Furthermore, one has the bound  sup  x∈K  sup  λ<1  sup  φ∈Br  sup  τ∈Tα \\{0}  |Πxτ(φλ  x)|  λα|τ|α  ≤ ∥Π∥α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' ¯K∥Γ∥α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' ¯K,  as well as the analogous difference bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Furthermore, one has the following simple consequence of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='31 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Under the assumptions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='30 one has for each α ∈ A \\△, τ ∈ Tα  and δ ∈ A one has  sup  x,y∈K:|y−1x|<1  |Γx,yIτ|δ+β  |τ|α−β|y−1x|α−δ ≲ (1+∥Γ∥α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' ¯K)∥Π∥α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' ¯K +  sup  x,y∈K:|y−1x|<1  |Γx,yτ|δ  |τ|α−β|y−1x|α−δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Again, the analogous bound for the difference of two models holds as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  4 APPLICATIONS TO SEMILINEAR EVOLUTION EQUATIONS  In this section we specialize to the setting, when the homogeneous Lie group G has distin-  guished time direction, see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4 below for illustrative examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Specifically, assume  we are given decomposition of the Lie algebra g = pc⊕p where both summands are s-invariant  subspaces and furthermore pc is one dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In line with this decomposition, in this  section we deviate from the convention stated above Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2) and reorder the basis of g  so that we always have the time component, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' X1 ∈ pc, in the first slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Under this assump-  tion we also fix the basis ¯X = {Xi}d  i=2 of p where sXi = si .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We note that P := exp(p) ⊂ G is a  homogeneous subgroup as in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3 and we recall the notation defined there,  ¯s := s|p  and  |¯s| := trace(s|p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  In particular we have |s| = s1 +|¯s|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' From now on we identify the Lie group G with R×P under  the diffeomorphism  R×P → G,  (t,x) �→ x exp(t X1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  and we shall often use the notation z = (t,x) ∈ R×P = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Let us collect some useful facts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Under this identification we see that R×{e} ⊂ G is a homogeneous subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  The scaling map behaves as expected, in that for any z = (t,x) ∈ G and δ > 0  δ·(t,x) = (δs1t,δ· x) ,  where δ· x is understood to be the restriction of the dilation to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  50  The map ˜Φ from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='20) is related to our decomposition here, in that for any X p ∈ p and  t ∈ R, we have ˜Φ(X p + t X1) = (t,exp(X p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Recall the map NP : G → R+ defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' There exists a constant c′ > 0 such that  NP(t,x) = c′|t|  1  s1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In particular, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='24) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='25) there exist a constant c > 0 such that  for any (t,x) ∈ G  1  c |t|  1  s1 ≤ d ((t,x),P) ≤ c|t|  1  s1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1)  A core assumption for the remainder of this section will be that of non-anticipativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We say that G : G \\{e} → R is non-anticipative if for any (t,x), (s, y) ∈ G one  has G((s, y)−1(t,x)) = 0 whenever s ≥ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  We will also require an assumption of prescribed homogeneity on the kernel, with respect  to the dilation map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For σ ∈ R, we say that that G : G\\{e} → R is smoothly σ-homogeneous if it is  smooth and for any z ∈ G\\{e} and λ > 0,  G(λ· z) = λσG(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  We now have the following analogue of [Hai14, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Given G : G\\{e} → R satisfying Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 and Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2 with σ = −|¯s|,  then there exists a smooth function ˆG : P → R such that for all (t,x) ∈ G with t > 0,  G(t,x) = t− ¯s  s1 ˆG  �  t− 1  s1 · x  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2)  For every multi-index I ∈ Nd−1 and every n > 0 there exists a constantC > 0 such that uniformly  over x ∈ G,  | ¯X I ˆG(x)| ≤ C(1+|x|2)−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The proof follows exactly the same steps of [Hai14, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4] after replacing the usual  derivatives there with the vector fields ¯X = {Xi}d  i=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Let G : G\\{e} → R satisfy Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 and Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2 with σ = −|¯s|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Then,  for any r > 0, there exist smooth functions K : G \\ {e} → R and K−1 : G → R, both satisfying  Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 and such that G = K +K−1 and  K is compactly and satisfies Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='25 with β = s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  K−1 : G → R is globally smooth and such that X IK−1 ∈ L∞  loc(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='L1(P)) for any I ∈ Nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The proof readily adapts from [Hai14, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5 & Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='7] with the only real change  being to skip the last step in the proof of [Hai14, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='7] and instead using Faa di Bruno’s  formula along with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3) to obtain the claimed integrabillity of K−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Note that more careful analysis yields improved bounds on K−1, however, K−1 ∈  L∞  loc(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='L1(P)) suffices for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  51  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 SHORT TIME BEHAVIOUR OF KERNEL CONVOLUTIONS  Together, the following two results show, in our setting, that the lift of the kernel applied to a  modelled distribution, f , can be controlled on sets of the form OT := {z ∈ G : d(z,P) ≤ T } only  using information about f on the same set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Note that using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1) there exists a constant c > 0  such that [0,cT ]×P ⊂ OT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Hence, the corresponding notion of local solutions, described in  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3, corresponds to the usual one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  First, we introduce some final pieces of notation, let R+ : R×P → R be a map such that for  all x ∈ P one has R+(t,x) = 1 for t > 0 and R+(t,x) = 0 for t ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' From now on we shall also  use subspaces of the previously introduced Hölder spaces (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4), modelled distri-  butions (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4) and singular modelled distributions (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5) writing, for ex-  ample  ¯Cα(G) ⊂ Cα(G),  ¯Dγ  α ⊂ Dγ  α,  ¯Dγ,η  α,P ⊂ Dγ,η  α,P,  where we allow the set K in the relevant definitions to be any closed subset K ⊆ OT for some  T > 0 (in particular K is not necessarily compact).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We shall often write the corresponding  semi-norms for example as  ∥ · ∥α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='OT ∩K,  |||·|||γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='OT ∩K,  |||·|||γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='OT∩K ,  where in particular we allow K = OT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  With these notions at hand the proof of the next theorem adapts directly from the proof of  [Hai14, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Let γ > 0, T be a regularity structure and M := (Π,Γ) and ¯M = ( ¯Π, ¯Γ) models  and K : G\\{e} → R a non-anticipative kernel (see Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1) such that the assumptions of  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='30 are satisfied for some β > 0 and a sector V of regularity α > −s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Then, for every  T ∈ (0,1], and η > −s1 and for κ > 0 small enough  ������KγR+ f  ������  γ+β,(η∧α)+β−κ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='OT ≲ T κ/s1������f  ������  γ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='OT  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4)  ������KγR+ f ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' ¯KγR+ f  ������  γ+β,(η∧α)+β−κ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='OT ≲ T κ/s1  �������f ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' ¯f  ������  γ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='OT +  ������M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' ¯M  ������  γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='O2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5)  The constant in the first bound depends only on |||M|||γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='O2, while in the second it may also de-  pend on  ������f  ������  γ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='OT ∨  ������ ¯f  ������  γ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='OT and |||M|||γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='O2 ∨  ������ ¯M  ������  γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='O2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The proof follows almost ad verbatim the proof of [Hai14, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1], only using our  modified definition of the sets OT := {z ∈ G : dG(z,P) ≤ T }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In fact, given any closed set K ⊂ G the bounds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5) both hold if OT is  replaced on the left hand side by OT ∩K and on the right hand side by OT ∩ ¯K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' However, we  will not make use of this fact in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  While Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6 treats the lift of the singular part of the kernel the following lemma  shows that the application of the smooth remainder can be lifted into the polynomial regu-  larity structure in a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  52  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Let K−1 : G → R be a smooth, non-anticipative kernel on G, such that for any  I ∈ Nd the functions X IK−1(t, ·) are bounded in L1(P), locally uniformly in t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Then, under  the assumptions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6 one has the bound  ���  ���  ���PPPγ  (·)[K−1RMR+ f ]  ���  ���  ���  γ+β,¯η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='OT ≲ T  ������f  ������  γ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='OT  as well as the analogous difference bound  ���  ���  ���PPPγ  (·)[K−1RMR+ f ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='PPPγ  (·)[K−1R ¯MR+ ¯f ]  ���  ���  ���  γ+β,¯η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='OT ≲ T  �������f ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' ¯f  ������  γ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='OT +  ������M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' ¯M  ������  γ,O1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The argument adapts mutatis mutandis form the proof of [Hai14, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3], replacing  the compact support assumption therein by the integrability property of K−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2 INITIAL CONDITIONS  We will only consider evolution equations on domains without boundary so that our only  boundary data is the initial condition and proceed as in [Hai14, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Let u0 ∈ Cα(P) such that ∥u0∥Cα(P) < ∞ and T = (T,G) be a regularity structure  containing the polynomial regularity structure over G and let G be a kernel satisfying Assump-  tions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We set for t ̸= 0  G(u0)(t,x) :=  �  T×R  G((0, y)−1(t,x))u0(y)dy,  where we used the suggestive but only formal notation if α ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Then, for every γ > 0 the map  PPPγ  (·)[G(u0)] : G\\P → T belongs to Dγ,α  0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='P for every γ > (α∧0) and furthermore, for any T > 0, one  has  ���  ���  ���PPPγ  (·)[G(u0)]  ���  ���  ���  γ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='OT < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We first decompose the kernel as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4 and write  G(u0)(t,x) = K (u0)(t,x)+K−1(u0)(t,x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  For the summands of K (u0)(t,x), one can argue as in the proof of [Hai14, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5] or as in  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The desired bound for K−1(u0)(t,x) follows exactly as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3 AN EXAMPLE FIXED POINT THEOREM  At this point, we have all ingredients at hand to straightforwardly see that the very general  fixed point Theorem [Hai14, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='8] as well as the other parts of [Hai14, Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3]  adapt to the setting of homogeneous Lie Groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For the sake of conciseness and with the  primary example of Anderson-type models in mind, we refrain from presenting this material  in all generality and instead show an example fixed point theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' This result covers Ander-  son type equations, such as the one treated in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Recall here the notation introduced  at the start of this section, in particular the identity |s| = s1 +|¯s|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  53  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Let α ∈ (−s1,0], γ > −α, η + α > −s1 and G : G \\ {e} → R be a kernel satisfy-  ing Assumptions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2 with σ = −|¯s| and the decomposition G = K + K−1 as in Lemma  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Furthermore, let T = (T,G) be a regularity structure containing the polynomial regularity  structure, equipped with an abstract integration operator I of order s1 and containing an el-  ement Ξ ∈ Tα, where α = min A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Then, given a model M = (Π,Γ) which realises K for I and  satisfies the assumptions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='30 along with any v ∈ Dγ,η  0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='P,, the map  MT ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='v : Dγ,η  0  → Dγ,η  0  U �→ KγR+(UΞ)+Pγ  (·)[K−1R+(UΞ)]+ v,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6)  is well-defined and there exists a unique fixed point for some T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Furthermore, if ΠeΞ and v are G-periodic and the model is adapted to the action of G on the  group (as defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='7), then the unique fixed point is as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Crucially, the solution map depends continuously on the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Note that using that the abstract mild equation given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6) is linear in U, the  local time of existence T > 0 can be chosen independ of the initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Thus, given  suitable bounds on the model over a suitably large set of the form {z ∈ G : d(z,P) ≤ T +1}, by  restarting the equation one obtains existence of a fixed point for arbitrary T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We refer to  [HL18, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In practice we will usually take v = Pγ[G(u0)] so that by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='9 the con-  dition v ∈ Dγ,η  0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='P is satisfied provided u0 ∈ Cη(P), where η + α > −s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The notation K and K−1  is carried over from Section 3 and Section 4, in particular K is the lift of K , the compactly  supported, singular part of the fundamental solution, see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6 and K−1 the smooth  remainder, see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proof sketch of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The proof is a straightforward adaptation of the proof of [Hai14,  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='8] and follows by applying Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='23, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='8 to show  that the map M ¯T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='v has a unique fixed point in Dγ,η  0  for some T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Since α = min A, the  modelled distribution x �→ Ξ is an element of Dγ,γ  α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='P for any γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Assuming U ∈ Dγ,η  0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='P by  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='23 one therefore has,  UΞ ∈ Dγ+α,η+α  α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='P  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  So by using the singular Schauder estimate, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='34, along with the assumption (η+  α)∧α > −s1, one has  K(UΞ) ∈ Dγ+α+β,((η+α)∧α)+β  (α+β)∧0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='P  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Then, we see that  γ+α+β > γ,  η < η+α+β  and  (α+β)∧0 > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Hence, one finds Dγ+α+β,((η+α)∧α)+β  (α+β)∧0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='P  �→ Dγ,η  0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='P which concludes the proof that MT ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='v is a self-  mapping on Dγ,η  α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' By comparing MT ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='vU and MT ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='v ¯U for U, ¯U ∈ Dγ,η  0  , one then shows that  MT ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='v is a contraction for sufficiently small T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  From the steps outlined above one furthermore obtains the the claims of G-periodicity and  the continuous dependence of the solution on the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  54  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4 CONCRETE EXAMPLES OF DIFFERENTIAL OPERATORS AND KERNELS  The theory developed in the proceeding sections provides the analytic framework to treat  singular SPDEs using the tools of regularity structures where the linear part of the equation is  given by an operator L satisfying the assumptions of Folland’s theorem, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In this  section we present a non-exhaustive list of homogeneous Lie groups and linear differential  operators, L, to which our results apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Generically, these equations are of the form,  Lu = ∂tu − ¯Lu = F(u, ¯Xu,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=',ξ),  u|t=0 = u0,  where ξ is a suitable noise, the non-linearity is linear in the noise and only depends on lower  order derivatives of u than are contained in L which satisfies the assumptions of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1,  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' [Fol75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Setting 1 P is a stratified Lie Group (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5) with a basis {Xi }m  i=1 of W1, generating the  Lie Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We equip G := R×P with the trivial homogeneous Lie Group structure  G×G ∋  �  (t,x),(t′,x′)  �  �→ (t + t′,xx′)  with the extended scaling λ·(t,x) = (λ2t,λ·x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The heat type operator associated to the  sub-Laplacian  L = ∂t −  m  �  i=1  X 2  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  satisfies the assumptions of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' and it follows from [Fol75, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 & Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3]  that L is non-anticipative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We shall discuss a notable example of the setting, that of the  heat operator on the Heisenberg group, below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Setting 2 A generalisation of the above setting is to take P a homogeneous Lie Group and equip  G := R×P with the same structure as above, but this time define the scaling λ·(t,x) =  (λ2q t,λ· x) for q ≥ 2 and an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' A natural class of operators is given by  LQ = ∂t −Q(X1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=', Xm) ,  where Q is a polynomial of homogeneous degree 2q and such that LQ and L∗  Q are hy-  poelliptic and LQ is non-anticipative, see [Hai14, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' A notable example is the heat  type operator associated to the bi-sub-Laplacian on a stratified Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Setting 3 The homogeneous Lie Group G = R×P is equipped with a non-trivial Lie group struc-  ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In this case we look at differential operators of the form,  L = X0 −Q(X1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=', Xm) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  where X0 = ∂t + ¯X0 and the operators L, L∗ satisfy the criteria of Folland’s theorem  and L is non-anticipative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Examples are parabolic/restricted Hörmander operators,  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' [Hai16, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1] or [Bel95, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 3] and the kinetic Fokker–Planck operator fits into  this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  55  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 HEAT OPERATOR ON THE HEISENBERG GROUP  In the context of Setting 1 and Setting 2 we recall that the Heisenberg group, Hn, defined in  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4, is a stratified Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Identifying Hn = R2n ×R we recall that  Ai(x, y,z) = ∂xi + yi∂z,  Bi(x, y,z) = ∂yi − xi∂z,  C(x, y,z) = ∂z  are left-translation invariant vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' It is readily checked that the operator  L = ∂t −  n�  i=1  (A2  i +B2  i ),  is homogeneous with respect to the scaling,  λ·(t,x, y,z) := (λ2t,λx,λy,λ2z),  Applying [Fol75, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1] there exists a unique, fundamental solution K : Hn → R associated  to L which is smooth away from e and satisfies the assumptions of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' As a result our  framework allows for the study of semi-linear evolution equations of the form,  ∂tu −Lu = F(u, A1u,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=', Anu,B1u,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=',Bnu,ξ),  u|t=0 = u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2 MATRIX EXPONENTIAL GROUPS AND KOLMOGOROV TYPE OPERATORS  A wide class of operators fitting into Setting 3 above are the Kolmogorov (or K-type) operators  on R×Rn for n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The group structure is a matrix exponential group, as defined in Section  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Given two, rational, n ×n block matrices,  A =  \uf8eb  \uf8ec\uf8ed  A0  ···  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  0  ···  0  \uf8f6  \uf8f7\uf8f8,  B =  \uf8eb  \uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed  0  B1  0  ···  0  0  0  B2  ···  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  0  0  ···  0  Bk  0  0  ···  0  0  \uf8f6  \uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  with A0 constant, positive definite and of rank q ≤ n and each Bi a pi−1 × pi block matrix of  rank pi, where q = p0 ≥ p1 ≥ ··· ≥ pk and �k  i=1 pi = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Then the linear operator, defined for  u : R×Rn → R by  Lu(t,z) = ∂tu(t,z)−∇·(A∇u(t,z))+Bz ·∇u(t,z),  satisfies Hörmander’s condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Equipping R×Rn with the matrix exponential group struc-  ture associated to B (and defined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4) it is readily checked that L satisfies the as-  sumptions of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In fact, there is an explicit formula for the fundamental solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  We first let  C(t) := 1  t  �t  0  exp(−sB⊤)A exp(−sB)ds  and then using the same notation for the effective spatial dimension, |¯s| := �k  i=0(2i +1)pi,  K (t,z) =  �  1  (4π)nt|¯s| detC(t)  �1/2  exp  �  −C(t)−1z · z  4t  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  56  The kinetic Fokker–Plank operator falls into this class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Consider the domain R × R2d, with  variables (t,x,v) and set, as block matrices,  A =  �Id  0  0  0  �  ,  B =  �0  Id  0  0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Then the associated operator is  Lu(t,v,x)= ∂tu(t,v,x)−∆vu(t,v,x)− v ·∇xu(t,v,x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  and the fundamental solution in fact has the explicit form,  K (t,v,x) =  2  �  3  d(4π)d t2d+1 exp  �  −|v|2  4t −  3  ��x + v  2 t  ��2  t3  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  See [IK64, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 7] for a derivation in the case d = 1 and [Man97] in the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  5 A REGULARITY STRUCTURE FOR ANDERSON EQUATIONS  In this section we present a brief construction of a sufficiently rich regularity structure T =  (T,G) in order to solve abstract fixed point equations of the form  U = I(UΞ)+U0 ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1)  as treated by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='10, where I denotes the abstract part of the lift of a 2 regularising  kernel as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5, Ξ is the lift of a noise of regularity α and U0 is the polynomial part of  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In order for the equation to be subcritical one needs to impose α > −2 and thus it follows  from the previous discussions that we can look for a fixed point in a space Dγ  P for γ < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Thus  we only need to incorporate abstract polynomials ηηηi with si < 2 into our regularity structure  and since one has the following identities in this case  ηi(xy) = ηi(x)+ηi (y),  Pγ  x[f ](·) = f (x)+  �  i :si<γ  ηi(·)Xi f (x)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2)  one can work with essentially a truncated version algebraic framework as on the abelian  group Rd developed in [BHZ19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Therefore, in the remainder of this short section, we freely  use notations and definitions from [BHZ19], often without further explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  We define two edge types L = {t,Ξ} and declare |t| = 2 and |Ξ| = α and as well as a scaling  in the sense of [BHZ19] s = (s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=',sn), where n = max{i ≤ d : si < 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 5 Motivated by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1) we  define the naive (normal) rule, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' [BHZ19, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='7]  ˚R(t) = {(t,Ξ),(t),(Ξ),()},  ˚R(Ξ) = {()}  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3)  and consider its completion R constructed in [BHZ19, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='21] (note that this second  step is only necessary if α ≤ −3/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We denote by T ex and T ex  + ,T− ⊂ T ex  −  the corresponding  spaces constructed in [BHZ19, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='26, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='29, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='22] and summarise some important  properties of these spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  5Recall form Section 2 that s1, s2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='.,sn are the first n eigenvalues of the scaling s in increasing order and that Xi  are the corresponding eigenvectors  57  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The triple {T ex,T ex  + ,T−} have the following algebraic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  The spaces T ex  + ,T−,T ex  −  are graded Hopf algebras with coproduct given by △+ and △−  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  The space T ex is a right comodule over T ex  + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  The spaces T ex and T ex  +  are left comodules over T− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  These spaces satisfy the co-interaction property, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' the following diagram commutes  T ex  T ex ⊗T ex  +  T ex  − ⊗T ex  T ex  − ⊗T ex ⊗T ex  +  △+  △−  M(1,3)(2)(4)◦(△−⊗△−)  id⊗△+  Note that the fact that we can leverage the framework developed in [BHZ19] relies crucially  on the fact that it suffices to include polynomials of degree < 2, which translates into the  algebraic relation △+(ηηηj) =ηηηj ⊗1+1⊗ηηηj whenever sj < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In general one would need a mod-  ification ˜△+ of the map △+ such that ˜△+(ηηηj) = �  d(I)+d(J)=sj C I,J  j ηηηI ⊗ηηηJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Furthermore, one  would need a modification ˜△− of triangle △− and both modifications, ˜△+ and ˜△−, would  depend on the group structure of G, since the form of higher order Taylor polynomials de-  pends crucially on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' To circumvent these considerations we restrict ourselves to working  with the following subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Let T consists of the span of those basis vectors (trees) T ∈ T ex where T and its grading |T |+,  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' [BHZ19, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3], satisfies the following properties  it is planted and satisfies |T |+ < γ or T = T ′Ξ where T ′ is planted and satisfies |T ′|+ < γ,  the only polynomial decorations appearing are at the second highest node and of de-  gree < γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Similarly, we set T+ ⊂ T ex  +  to consist of the subalgebra generated by those trees T ∈ T ex  + satis-  fying  |T |+ < γ  the only polynomial decorations appearing are at the second highest node and of de-  gree < γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Observe that Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 still holds with T ex  +  replaced by T+ and T ex replaced by T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We  define G+ to be the character group of T+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' From now on our regularity structure shall be given  by  T = (T,G) ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4)  where G consists of the elements Γ ∈ L(T,T ) of the form Γ = (id⊗ f )△+ for some f ∈ G+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  58  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 SMOOTH MODELS  In order to define the models, we have to deviate slightly from [BHZ19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For i ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=',n} we  write ηηηi instead of Xi for abstract polynomials and we declare a map ΠΠΠ : T → C ∞(G) (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  [BHZ19, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='8]) to be admissible for a smooth function ξ ∈ C ∞(G) and a kernel K as in  Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='25, if for every x ∈ G  ΠΠΠΞ(x) = ξ(x),  Πηηηi(x) = ηi(x)  and  ΠΠΠIτ = KΠΠΠτ,  ΠΠΠIiτ = (XiK )ΠΠΠτ ,  where we write I resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Ii for the operation of attaching an edge of type t, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' (t,ei) to the  root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For ΠΠΠ an admissible map as above and z ∈ G we recursively define fz ∈ G+ and Πz by  fz(Iτ) = −  �  d(I)<|It(τ)|+  ηI (z−1)X IK (Πzτ)(z)  ΠzIτ(¯z) = K (Πzτ)(¯z) −  �  d(I)<|I(τ)|+  ηI(z−1 ¯z)X IK (Πzτ)(z) ,  and by the analogous expression for Iiτ, where i ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=',n}, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' [BHZ19, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We say  that ΠΠΠ is canonical, if it satisfies ΠΠΠΞτ = ΠΠΠΞΠΠΠτ for every τ such that Ξτ ∈ T , and it does not  see the extended decoration, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='e is reduced in the sense of [BHZ19, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' One can check  that for such canonical lifts ΠΠΠ the maps Πz = (ΠΠΠ⊗ fz)△+ and Γγx,y where γx,y = (f −1  x  ⊗ fx)△+  form a model  M = (Π,Γγ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5)  The renormalisation group G− is defined to be the character group of the Hopf algebra T ex  − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  We observe that in our case T ex  −  coincides (as an algebra) with the free algebra spanned by  a family of linear trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' (In the case α > −3/2, one furthermore has T ex  − = T− and we list the  trees explicitly in the next section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The group G− acts on models as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For g ∈ G− let  Rg = (g ⊗id)△−, we set  ˆM g = ( ˆΠg, ˆΓg) = (ΠRg,ΓγRg ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6)  One can check that every element the orbit of a smooth canonical model as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5) is indeed  a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Note that in order to show convergence of a family of models ˆM(ε) = ( ˆΠ(ε), ˆΓ(ε))  of the form (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6) for the regularity structure T = (T,G) for some smooth underlying noises  ξǫ and a sequence of elements gǫ ∈ G−, it is sufficient to show convergence of ˆΠε|T≤0 due to  Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='36 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  We emphasise again that this section relied on the the assumption γ < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  6 THE ANDERSON EQUATION ON STRATIFIED LIE GROUPS  In this section we apply the machinery developed in this article in order to solve a class of  parabolic Anderson type models on a compact quotients of an arbitrary d-dimensional, strat-  ified, Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Let G be a stratified Lie group (recall Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5) with Lie algebra g and  59  G a lattice subgroup with its canonical left action on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Recall from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3 the definition  of the quotient map π : G → S = G/G, we shall, without loss of generality assume that there  exists a fundamental domain of this action, K ⊂ G, such that e ∈ BN(e) ⊂ K for some N ∈ N,  large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6 For m ≤ d, let X1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=', Xm be a basis of W1, the family of left invariant vector  fields generating g and we write ˜Xi = π∗Xi for the push-forward vector fields on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Finally we  recall the notion of convolution ∗S on S induced by the right action of G on S, as defined in  Section (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  We shall solve equations where the driving noise satisfies the following assumption for  some ¯α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Assumption 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For ¯α ∈ (−|s|/2,0), assume that ξα is of the form ξ = ˜ξ∗S c ¯α, where ˜ξ denotes a  Gaussian white noise7 on S and c ¯α : G\\{e} → R is a function with bounded support, satisfying  |c ¯α(x)| ≲ |x|−|s|+(|s|/2+¯α) as well as c ¯α(x−1) = c ¯α(x) for all x ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For ξ satisfying Assumption 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1, a simple calculation shows that for any φ,ψ ∈  L2(S),  E[〈ξ,φ〉〈ξ,ψ〉] =  �  S  (φ∗S c ¯α)(x)(ψ∗S c ¯α)(x)dx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  In particular, coloured noise where the regularisation is given by a negative fractional power  of the sub-Laplacian fits into our setting, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' [BOTW22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The assumption that c ¯α(z) = c ¯α(z−1) is not crucial, but allow us to reuse com-  putations previously carried out for equations on Euclidean domains in [HP15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' A robust  treatment of similar equations in the Euclidean setting, driven by more general driving noise,  even beyond the Gaussian setting, is given in [CS17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' A similar remark holds for the choice of  mollifier ρ in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  In the above setting, we have the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Fix T > 0 arbitrary and let ξ be a noise satisfying Assumption 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 with ¯α ∈  (−3/2,−1) and η ∈ (−(2 + ¯α),0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For ε ∈ (0,1), let uε : [0,T ] × S → R be the unique solution  to the random PDE  ∂tuε =  m  �  i=1  ˜X 2  i uε +uε(ξε −cε),  u|t=0 = u0 ∈ Cη(S) ,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1)  where ξε := ξα ∗S ρε, for ρ ∈ C∞  c (B1(e)) satisfying ρ(z) = ρ(z−1) and {cε(ρ)}ε∈(0,1) is a family of  (diverging) constants, depending on ρ and such that |cε| ≲ ε2+2 ¯α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Then, there exists a random  function u : [0,T ]×S → R, independent of the chosen mollifier, ρ ∈ C∞  c (B1(e)), such that  sup  (t,x)∈[0,T ]×S  t−η|uε(t,x)−u(t,x)| → 0  in probability as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  6This is for example achieved by replacing the homogeneous norm on G with a multiple of itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We only require  this condition in order to make the integrands in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 supported on the fundamental domain K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  7Recalling that S is a measure space, ˜ξ can be defined as a centred Gaussian field, over L2(S) and with covariance  given by the L2(S) inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  60  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We point out that the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4 given below modifies mutatis mu-  tandis to the case where the noise is allowed to depend on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In this case the natural mod-  ification of Assumption 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 is to assume that the noise is of the form ξ = ˜ξ∗c ¯α, where ˜ξ is a  space-time white noise on R×S and c ¯α satisfies |c ¯α(t,x)| ≤ (|x|+|t|1/2)−|s|/2−1+ ¯α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The only dif-  ference in the proof is to replace all convolutions and integrals over S with convolutions and  integrals over R×S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We point out that this form of noise is not covered by [BOTW22], where  the white in time assumption is required in order to make use of martingale arguments, see  also Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='7 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The range of ¯α considered in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4 does not cover the full subcritical  regime of the Anderson equation on a stratified Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Indeed one expects an analogue of  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4 to hold for any ¯α ∈ (−2,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The main obstacle is the absence of a BPHZ type the-  orem in the setting of homogeneous Lie groups, see [CH16, HSb] for the corresponding result  on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Let us further point out that in order for the Carnot group G to be non-trivial it must  have scaled dimension |s| ≥ 4, hence the sub-critical regime for ¯α is necessarily contained in  Assumption 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Let us point to the work [BOTW22], where, using Itô calculus methods, the au-  thors construct a solution-theory in the case, when the noise is white in time and coloured in  space, corresponding to the full subcritical regime on the Heisenberg group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' This probabilis-  tic notion of solution circumvents the need for explicit renormalisation as in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Further-  more, their results are obtained on the infinite volume group Hn, rather than the compact  quotient space that we consider here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We expect that by using weighted modelled distribu-  tions, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' [HL18], together with the techniques developed in [HP15], one can recover the  solution constructed in [BOTW22] together with a Wong–Zakai type theorem in the analogue  of the regime ¯α ∈ (−3/2,−1), treated by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' If one is interested in the full subcriti-  cal regime however, this would not just require a suitable BPHZ-type theorem in the setting  of homogeneous Lie groups, but furthermore a systematic way to single out the Itô model  among an (arbitrarily) high dimensional family of models obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We follow a usual strategy in the theory of regularity structures, show-  ing the equivalent theorem for the pulled back equations on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For this, we work with the  regularity structure T = (T,G) constructed in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4) with |Ξ| = (−3/2, ¯α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Note that since we are  in Setting 1 we can directly apply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 to see that there exists a unique, fundamental  solution G : (R×G)\\(0,e) → R and satisfying the assumptions of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4, so that we have  G = K +K−1 enjoying the properties described therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Denote by M(ε) = (Π(ε),Γε) the canonical model for T , satisfying Π(ε)  z Ξ = ξε and realising  K for I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' From now on we shall use the common tree notation, with the obvious changes in  interpretation, as defined for example in [HP15, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1], and write  := Ξ,  = IΞ,  = ΞIΞ,  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Since ¯α ∈ (−3/2,−1) the rule defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3) is complete and T− coincides (as an algebra)  with the free commutative algebra generated by  �  ,  ,  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Since the noise is Gaussian and  centred, it is sufficient to work with the subgroup ˜G− ⊂ G− consisting of g ∈ G− satisfying  g(τ) = 0  ⇒  τ ∈ span  �  ,  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  61  For each ε ∈ (0,1) we fix gε ∈ ˜G− to be the character specified by  gε(  ) = E[ξε(e)K (ξε)(e)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Thus, applying Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='10, we obtain a solution Uε : [0,T ]×G → T<γ to the abstract lifted  equation, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='6), for the model ˆM(ε) := M gε  ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Next we show that uR  ε := R ˆM(ε)Uε actually solves  Equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1) in several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  First, note that for any z = (t,x) ∈ [0,T ]×G, the abstract solution has the explicit form  Uε(z) = u0  ε(z)  �  1+  +  �  +  �  si<γ  ui  ε(z)ηηηi ,  since we are working with the expansion up to order γ < 3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' In particular ˆΠ(ε)  z Uε(z)(z) =  u0  ε(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  We observe that therefore  (Uε )(z) = u0  ε(z)  �  +  +  �  +  �  si<γ  ui  ε(z)ηηηi  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  We also note that ˆΠ(ε)  z  ��  si<γui  ε(z)ηηηi  �  (z) = 0 and  ˆΠ(ε)  z  �  u0  ε(z)  �  +  +  ��  (z) = u0  ε(z)  �  ˆΠ(ε)  z  (z)+ ˆΠ(ε)  z  �  +  �  (z)  �  = u0  ε(z)(ξε(z)−cε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Thus, we can conclude using Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='19 that  R ˆM(ε)(U g ) = u0  ε(z)(ξε(z)−cε) =  �R ˆM(ε)U g �  (z)(ξε(z)−cε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Thus it only remains to show is that the family of models ˆM(ε) converges to a limiting model  M, which is the content of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='8, stated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The familly of models ˆM(ε) = ( ˆΠ(ε), ˆΓε) constructed in the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4  converges as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Note that by Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2 it is sufficient to show convergence of ˆΠ(ε)|T<0 in the model  topology, which follows from a stright-forward adaptation of Kolmogorov’s criterion to mod-  els using the molifiers ϕ(n) constructed in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='10 together with the estimates obtained  in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For a detailed proof of a Kolmogorov type criterion for general models on Eu-  clidean domains, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' G = Rd, in the spirit above, we refer to [HSb, Appendix B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  62  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1 STOCHASTIC ESTIMATES FOR THE RENORMALISED MODEL  In this subsection we obtain convergence the renormalised models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We shall freely use tech-  niques of [Hai14, Sec 10] and as well as graphical notiation similar to that in [HP15, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 5],  with some slight changes in interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For example we write  �  Π(ε)  (0,e)  �  (ϕλ) =  +  ,  with the following interpretations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  The node  represents the origin, (0,e) ∈ R×G while the edge  represents integra-  tion against the rescaled test function ϕλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  The node  represents an instance of the G left periodic white noise on G while the  edge  represents the kernel c ¯α ∗ρε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Note therefore, that when ε = 0 our diagrams  will still contain dotted lines, as opposed to [HP15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  The nodes  represent dummy variables z = (t,x) ∈ R × G which are to be integrated  out and the edges  represent integration against the kernel K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' A barred arrow  represents a factor of K (t − s, y−1x) − K (−s, y−1), where (s, y) and (t,x) are the  coordinates of the start and end points of the arrow respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  As in [Hai14, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2] and [HP15, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1] we will also use the notation W(ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='k)τ to denote  the kernel associated to the kth-homogeneous Wiener chaos component of Π(0,e)τ and sim-  ilarly ˆ  W(ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='k)τ for the renormalised model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' For example, we find that ˆW(ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2)  ((s, y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='x1,x2) =  W(ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2)  ((s, y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='x1,x2) is given by  W(ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2)  ((s, y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='x1,x2) :=  �  R×G  (c ¯α∗ρε)(y−1  1 x1)(c ¯α∗ρε)(y−1x2)  �  K (s − s1, y−1  1 y)−K (−s1, y−1  1 )  �  ds1dy1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Fix δ > 0 small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Then, for every λ ∈ (0,1), ϕ ∈ Br and z = (t,x) ∈  R×G, there exist random variables  ( ˆΠz )(ϕλ),  ( ˆΠz  )(ϕλ),  ( ˆΠz  )(ϕλ)  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2)  such that for any p ≥ 1 the following estimates hold uniformly over λ,ε ∈ (0,1)  1a) E  �  |( ˆΠ(ε)  z  )(ϕλ)|p�  ≲ λp( ¯α−δ),  2a) E  �  |( ˆΠ(ε)  z  )(ϕλ)|p�  ≲ λp(2+2 ¯α−δ),  3a) E  �  |( ˆΠ(ε)  z  )(ϕλ)|p�  ≲ λp(4+3 ¯α−δ),  1b) E  �  |( ˆΠ(ε)  z  )(ϕλ)−( ˆΠz )(ϕλ)|p�  ≲ εpδλp( ¯α−δ) ,  2b) E  �  |( ˆΠ(ε)  z  )(ϕλ)−( ˆΠz  )(ϕλ)|p�  ≲ εpδλp(2+2 ¯α−δ) ,  3b) E  �  |( ˆΠ(ε)  z  )(ϕλ)−( ˆΠz  )(ϕλ)|p�  ≲ εpδλp(4+3 ¯α−δ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Firstly, we note that by translation invariance of the noise it is enough to consider  z = (0,e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Furthermore, since all random variables belong to a finite Wiener chaos, it suffices  63  to show these estimates for p = 2, [Hai14, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The first two estimates, 1a) and 1b),  follow readily by Ito’s isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  For the next items we recall [Hai14, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='14 & 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='18], the natural analogues of which  hold in our setting as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We can write (with the adaptation of the meaning of the diagrams  described above) the Wiener chaos decomposition of the second symbol as  � ˆΠ(ε)  (0,e)  �  (ϕλ) =  −  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3)  The first summand is the graphical representation of the iterated integral, I2(( ˆ  W(ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2)  )(z))  integrated against a test function centred at (0,e) ∈ R×G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' see for example the analogous sec-  ond term in [Hai14, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='23)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The second summand is the graphical analogue of the first  term in [Hai14, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='23)] given by I0(( ˆ  W(ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='0)  )(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Next note that by virtually an identical  calculation as in the beginning of the proof of [Hai14, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='19] one finds,  |〈(( ˆ  W(ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1) )(z),( ˆ  W(ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='1) )(¯z)〉| ≲ |z|2 ¯α+4 +|¯z|2 ¯α+4 ,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4)  which implies  |〈( ˆW(ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2)  )(z),( ˆ  W(ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2)  )(¯z)〉| ≲ (|z|2 ¯α+4 +|¯z|2 ¯α+4)|¯z−1z|2 ¯α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Together with the simple estimate I0(( ˆ  W(ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='0)  )(z))| ≲ |z|2 ¯α+2 this gives 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Next we turn to show 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' First, note that by a similar calculation as for (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='4) one finds  |〈( ˆW(ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2)  )(w),( ˆW(ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2)  )( ¯w)〉| ≲ |w|2(4+2 ¯α) +| ¯w|2(4+2 ¯α) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Therefore, together with the bound I0(( ˆ  W(ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='0)  )(w))| ≲ |w|2 ¯α+4 we find  E  ���  �  ( ˆΠ(ε)  (0,e)  )⋄( ˆΠ(ε)  (0,e) )  �  (ϕλ)  ���  2 ≲ λ2(4+3 ¯α) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  As in [HP15, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2] we find  �  ˆΠ(ε)  (0,e)  �  (ϕλ) =  +  \uf8eb  \uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed  −  \uf8f6  \uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8  +  −  and introduce the notationQε(s, y) = K (s, y)(cα∗ρε)∗2(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' We see that cε =  �  R×GQε(z)dz, and  define the renormalised kernel RQε(s, y) := Qε(s, y)−cεδ(0,e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Then, as in [HP15, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='14)],  using the notation  for the renormalised kernel, one finds  �  ˆΠ(ε)  (0,e)  �  (ϕλ) =  +  −  +  −  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  64  Similarly,  �  ( ˆΠ(ε)  (0,e)  )⋄( ˆΠ(ε)  (0,e) )  �  (ϕλ) =  −  and thus it only remains the bound the covariance of the diagrams  ,  ,  ,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='5)  The latter two can be bounded by repeatedly using the analogue of [Hai14, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='14], while  for the first diagram we use additionally the natural analogue of [Hai14, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' This  concludes the proofs of 3a) and therefore the first column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The bounds on the random vari-  ables in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='2) are obtained analogously by replacing c ¯α ∗ρε by just c ¯α throughout (and appro-  priately interpreting the renormalised kernel, Qε, for ε = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' The approximation bounds 2b)  and 3b) are obtained by standard arguments, replacing each diagram with a finite telescopic  sum of diagrams, where all dotted lines except one either encode convolution with c ¯α ∗ρε or  convolution with c ¯α and the one remaining dotted line is replaced by c ¯α −c ¯α ∗ ρε and then  applying the analogue of [Hai14, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
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+page_content=' Bernicot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  High order paracontrolled calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Forum Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
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+page_content=' Bailleul and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Bruned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Locality for singular stochastic PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='00399,  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  [BCCH21] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
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+page_content=' Hairer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Renormalising SPDEs in regu-  larity structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
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+page_content='  65  [BDM22]  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Bailleul, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
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+page_content=' Mouzard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Analysis of the Anderson operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
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+page_content='04705, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
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+page_content=' Degenerate stochastic differential equations and hypoellipticity, vol-  ume 79 of Pitman Monographs and Surveys in Pure and Applied Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Longman, Harlow, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
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+page_content=' Hairer, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Zambotti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Algebraic renormalisation of regularity  structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
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+page_content=' Bailleul and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Mouzard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Paracontrolled calculus for quasilinear singular PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  Stochastics and Partial Differential Equations: Analysis and Computations, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
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+page_content=' Ouyang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
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+page_content='  An invitation to hypoelliptic operators and Hörmander’s vector  fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Springer Briefs in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Springer, Cham, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content='  [CH16]  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Chandra and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
+page_content=' Hairer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'}
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