diff --git "a/INE4T4oBgHgl3EQfhA00/content/tmp_files/2301.05121v1.pdf.txt" "b/INE4T4oBgHgl3EQfhA00/content/tmp_files/2301.05121v1.pdf.txt" new file mode 100644--- /dev/null +++ "b/INE4T4oBgHgl3EQfhA00/content/tmp_files/2301.05121v1.pdf.txt" @@ -0,0 +1,4696 @@ +arXiv:2301.05121v1 [math.PR] 12 Jan 2023 +Singular SPDEs on Homogeneous Lie Groups +Avi Mayorcas1 and Harprit Singh2 +1Institute of Mathematics, Technische Universität Berlin, Str. des 17. Juni 136, 10587 Berlin, Germany. +Email: avimayorcas@gmail.com; ORCID iD: 0000-0003-4133-9740 +2Department of Mathematics, Imperial College London, South Kensington Campus, SW7 2AZ, UK. +Email: h.singh19@imperial.ac.uk; 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. 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. Prime examples +are the kinetic Fokker-Planck operator ∂t − ∆v − v · ∇x and heat-type operators associ- +ated to sub-Laplacians. 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. +Keywords: Regularity structures, homogeneous Lie groups, hypoelliptic operators, stochastic +partial differential equations. +2020 MSC: 60L30 (Primary); 60H17, 35H10, 35K70 (Secondary) +CONTENTS +1 Introduction +2 +1.1 +Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +1 + +1.2 +A Motivating Class of Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +1.3 +Open Problems and Wider Context +. . . . . . . . . . . . . . . . . . . . . . . . . . +7 +2 Analysis on Homogeneous Lie Groups +9 +2.1 +Derivatives and Polynomials +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +2.2 +Distributions and Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +17 +2.3 +Discrete Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +19 +2.4 +Concrete Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +20 +3 Regularity Structures and Models +22 +3.1 +The Polynomial Regularity Structure . . . . . . . . . . . . . . . . . . . . . . . . . . +23 +3.2 +Modelled Distributions +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +25 +3.3 +Singular Modelled Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . +33 +3.4 +Local Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +38 +3.5 +Convolution with Singular Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . +40 +3.6 +Schauder Estimates for Singular Modelled Distributions . . . . . . . . . . . . . . +48 +3.7 +Symmetries +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +49 +3.8 +Bounds on Models +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +50 +4 Applications to Semilinear Evolution Equations +50 +4.1 +Short Time Behaviour of Kernel Convolutions . . . . . . . . . . . . . . . . . . . . +52 +4.2 +Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +53 +4.3 +An Example Fixed Point Theorem +. . . . . . . . . . . . . . . . . . . . . . . . . . . +53 +4.4 +Concrete Examples of Differential Operators and Kernels . . . . . . . . . . . . . +55 +5 A Regularity Structure for Anderson Equations +57 +5.1 +Smooth Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +59 +6 The Anderson Equation on Stratified Lie Groups +59 +6.1 +Stochastic Estimates for the Renormalised Model . . . . . . . . . . . . . . . . . . +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) , +(1.1) +when the operator L is uniformly elliptic. 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. 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. +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. +2 + +1. The first, analytic step, is to introduce the notions of a regularity structure, models and +modelled distributions; generalised Taylor expansions of both functions and distribu- +tions. 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. Cru- +cially, a reconstruction operator maps modelled distributions to genuine distributions +(or functions) on the underlying domain. 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]. +2. 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]. In [BHZ19] the au- +thors construct concrete (equation-dependent) regularity structures and models to- +gether with a large enough renormalisation group. Then, in [BCCH21], the question +of how this renormalisation group acts on the SPDE in question was answered. Con- +vergence of renormalised models for an extremely large class of noises, known as the +BPHZ theorem, was proved in [CH16]. +In much of the above work translation invariance on Rd, of both the equation and the driving +noise, plays a major role. 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). +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. 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. 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. 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. +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. 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. +(1.2) +We will carry these two operators and their associated homogeneous Lie groups, throughout +the paper as working examples of the theory. 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. +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. +Theorem 1.1 ([Fol75, Thm. 2.1 & Cor. 2.8]). 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. 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. +where the convolution is with respect to the given Lie structure. +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) . +The following celebrated result of Hörmander (almost) entirely classifies the family of second +order hypoelliptic operators. +Theorem 1.2 ([Hö67, Thm. 1.1]). Let r ≤ d and {Xi }r +i=0 be a collection of first order differential +operators (i.e. vector fields) on Rd and recursively define V1 = span{Xi : i = 0,...,r}, Vn+1 = +Vn ∪{[V,W ] : V ∈ V1, W ∈ Vn}. If a differential operator can be written in the form, +D = +r� +i=1 +X 2 +i + X0 +c, +(1.3) +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. +Remark 1.3. By Froebenious’s theorem, [Fro77], if the condition of Theorem 1.2 fails in some +open set then D is not hypoelliptic on that set. However, the statement is not truly sharp; +for example the Grushin operator ∂2 +vv − v∂x is hypoelliptic, while the conditions of Theo- +rem 1.2 are violated on sets intersecting {v = 0}. 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. +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. +For example, it gives no direct route to a refined regularity theory for hypoelliptic equations. +This observation highlights the contribution of Folland’s theorem (Theorem 1.1); since trans- +lation invariance allows access to many additional tools in the Euclidean setting, such as +harmonic analysis and singular integral methods. 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. +We refer to [Bra14, Ch. 3] for a concise introduction to this programme. +1.1 RELATED LITERATURE +Non-translation invariant and non-uniformly elliptic SPDEs have been well-studied in the +classical, i.e. non-singular, regime and we do not attempt to present this large literature here. +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. 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. A solution theory for conservative SPDEs based on the kinetic +Fokker–Planck equation (see Section 4.4) in the Itô case was developed in [FFPV17]. Using +the theory of paracontrolled calculus this was extended to the singular regime in [HZZZ21]. +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. 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. We treat a closely related problem in the final sections of this paper. A fuller +discussion of the similarities and differences between the results of [BOTW22] and those of +our approach is postponed to Remark 6.7. +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]. Some of these works are discussed in more detail in Section 1.3 +below. Quasilinear SPDEs have been studied using regularity structures, rough path based +methods and paracontrolled distribution theory, see [GH19b, OW19, BM22]. Recently, an ap- +proach inspired by the theory of regularity structures, but technically distinct, has been devel- +oped in [OSSW21, LOT21, LO22]. SPDEs on domains with boundaries have been treated us- +ing both regularity structures and paracontrolled distribution based methods, [Lab19, GH19a, +CvZ21, GH21]. Finally, a number of works have considered parabolic, singular SPDEs on Rie- +mannian manifolds; a paracontrolled approach using the spectral decomposition associated +to the Laplace–Beltrami operator has been developed in [BB16, BB19, Ant22]. 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. 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. +1.2 A MOTIVATING CLASS OF EXAMPLES +In this paper we restrict ourselves to linear operators satisfying the criteria of Theorem 1.1 +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]. The parabolic Anderson +model +∂tu = ∆u −uξ, +u|t=0 = u0, +(1.4) +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]. 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]. 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]. +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 1.2), 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 . +(1.5) +When realisations of the environment are sufficiently singular then a pathwise solution the- +ory for (1.5) is expected to require renormalisation. 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 (1.5). 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). +We show a result analogous to those of [Hai14, HL18], finding a notion of renormalised so- +lution to (1.5), when ξ is a coloured periodic noise on a stratified Lie group. 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 (1.5) converge in probability to a unique limit +independent of the specific choice of mollification-scheme. We stress that this result does +not cover the full subcritical regime of (1.5). The treatment of this full regime would require +an analogue of the BPHZ theorem on homogeneous Lie groups, see [CH16, HSb]. +A notable example of a stratified Lie group is the Heisenberg group, Hn ∼= R2n ×R, see Sec- +tion 2.4.1 for a description. 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,...,n. +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 2.4.1 and that one has C(x, y,z) := [Ai,Bi] = −2∂z for +all i = 1,...,n. 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 (1.2) and (1.5), c.f. Section 4.4.1 for further +discussion. 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]. 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.g. [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. The solution theory exposited in Section 6 applies to precisely this example on a +compact quotient of the Heisenberg group. +1.3 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. 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]. This parallel progression in PDE analysis +leads one to ask, how far the theory of regularity structures can be extended in these direc- +tions. 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. The two rows describe the two parallel progressions outlined above in the con- +text of regularity structures. +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. +Mostly understood: analytic +step in [Hai14]; aspects of +renormalisation addressed in +[BB21]. +In [DDD19] 2d-PAM is treated. +A general account in +forthcoming work [HSa]. +Translation invariant +operators on homogeneous +Lie groups +General Hörmander +operators +On sub-Riemannian +manifolds +Analytic theory covered in this +work. Renormalisation and +convergence by hand. +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. +• Open Problem A is expected to be more involved than its counterpart in the Euclidean +setting; going from translation invariant operators to non-translation invariant oper- +ators by local approximations. In the case of general Hörmander operators, the un- +derlying Lie group structure would in general vary from point to point. An interesting +application would be the study of general hypoelliptic Anderson models (1.5) where +the underling particles perform a general diffusion with generator of the form (1.3), c.f. +Theorem 1.2. +• 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.f. [ABB20]. +Note that in view of Remark 1.3 the difficulties in Problem B are expected to be some- +what complementary to those of Problem A. +NOTATION: +Given α ∈ R we let [α] := max{r ∈ Z : r ≤ α} denote the integer part of α. 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. 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. We will use | · | to denote +the size of various quantities; 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. 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. For +integrals we will use the standard notation, dx, for integration against the Haar measure on a +given homogeneous, Lie group, see Proposition 2.1. Throughout most of the article we take +an intrinsic point of view and do not equip the homogeneous Lie group with an explicit chart. +STRUCTURE OF THE PAPER: +We begin with a discussion of analysis on homogeneous Lie +groups sufficient for our purpose, Section 2. The most important result in this section is The- +orem 2.13 which provides us with an intrinsic version of Taylor’s theorem on homogeneous +Lie groups and will be used frequently throughout. 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. 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. +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. 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. 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. +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. +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. +HS acknowledges funding by the Imperial College London President’s PhD Scholarship. He +wishes to thank Josef Teichmann and Máté Gerencsér for their hospitality during his visit to +ETH Zürich and TU Wien. +8 + +2 ANALYSIS ON HOMOGENEOUS LIE GROUPS +We collect some basic facts on homogeneous Lie groups and smooth function on them. Let +g be a Lie algebra and G be the unique, up to Lie algebra isomorphism, corresponding sim- +ply connected Lie group. We write [·,·] for its Lie bracket and inductively set, g(1) = g and +g(n) = [g(n−1),g]. Recalling the exponential map exp : g → G, we have the Baker–Campbell– +Hausdorff formula +exp(X )exp(Y ) = exp(H(X ,Y )) , +(2.1) +where H is given by H(X ,Y ) = X + Y + 1 +2[X ,Y ] + ... with the remaining terms consisting of +higher order iterated commutators of X and Y ; crucially H is universal, i.e. it does not depend +on the underlying Lie algebra. +Proposition 2.1 ([FS82, Prop. 1.2]). Assume that g is nilpotent, i.e. g(n) = 0 for some n ∈ N. +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). +• the pull-back to G of the Lebesgue measure on g is a bi-invariant Haar measure on G. +Definition 2.1. 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. +Remark 2.2. The requirement that the smallest eigenvalue of s be 1 is purely cosmetic. Oth- +erwise, denoting the smallest eigenvalue by s1, one can work with the new operator ˜s = 1 +s1 s. +Remark 2.3. For a ∈ R, denote by Wa ⊂ g the eigenspace of s with eigenvalue a. 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 . +In particular, if g admits a family of dilations, it is nilpotent. The converse is not necessarily +true. +Definition 2.2. 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. For r > 0, we define the group +automorphism +x �→ r · x := exp◦Dr ◦exp−1 x . +A homogeneous norm on G is a continuous function |·| : G → [0,∞) satisfying the following +properties for all x ∈ G, r ∈ R +9 + +1. |x| = 0 if and only if x = e, the neutral element, +2. |x| = |x−1| , +3. |r · x| = r|x| . +The homogeneous norm naturally induces a topology generated by the open sets and in +turn a Borel σ-algebra. From now on, we will always assume that G is equipped with this +topology and σ-algebra. 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 ≤ ... ≤ sd and such that +sX j = sj X j . +(2.2) +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.f. Proposition 2.1. In +integrals we use the standard notation dx. 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| . +We also define balls of radius r > 0, +Br(x) := {y ∈ G : |x−1y| < r} . +The topology induced by these balls agrees with the topology of G as a Lie group, see [FR16, +Sec. 3.1.6]. Note that due to the non-commutativity of G, in general |x−1y| ̸= |yx−1|. We +consistently work with the following choice a semi-metric on the group, +dG(x, y) := |x−1y| = |y−1x| . +For K ⊂ G we write ¯K := {z ∈ G : dG(z,K) := infy∈K|y−1z| ≤ 1} for the 1-fattening of K. +Remark 2.4. A function f : G → R is called homogeneous of degree λ ∈ R, if f (r · x) = r λf (x) +for all x ∈ G. One can show, c.f. [FS82, Prop. 1.5 & Prop. 1.6], 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| . +Furthermore all homogeneous norms are mutually equivalent and we may always choose a +homogeneous norm that is smooth away from e ∈ G, [FR16, Sec. 3.1.6]. +2.1 DERIVATIVES AND POLYNOMIALS +We identify g with the left invariant vector fields gL on G and write gR for the right invariant +vector fields. We write Xi for the the basis elements as in (2.2) seen as elements of gL and Yi +for the basis of gR satisfying Yi|e = Xi|e . 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). +A map P : G → R is called a polynomial if P ◦ exp : g → R is a polynomial on g.1 Let ζi be +the basis dual to the basis Xi of g. We set η j = ζj ◦ exp−1, which maps G to R. Note that +η = (η1,...,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,...,id) ∈ Nd +we write ηI = ηi1 +1 ·...·ηid +d . 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. 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}. 3 We can rewrite the group law on G explicitly in terms of η = (η1,...,ηd). +Proposition 2.5. For j ∈ {1,...,d} and multi-indices I, J s.t. 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) . +Proof. By the Baker–Campbell–Hausdorff formula (2.1) one has +η j(xy) = η j(x)+η j(y)+ +� +|I|+|J|≥2 +C I,J +j ηI (x)ηJ(y) . +By setting either x = e or y = e, we find that C I,J +j += 0 if I = 0 or J = 0. Since furthermore +η j((r x)(r y)) = r sj η j(xy) the claim follows. +Remark 2.6. Proposition 2.5 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) . +Remark 2.7. It is also noteworthy to realise that Proposition 2.5 implies that Pa is invariant +under right and left-translations. (This is not true if one replaces homogeneous degree by +isotropic degree, except if G is abelian or a = 0). +Remark 2.8. If follows from Proposition 2.5 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 . +1Recall that the space of polynomial functions on g is canonically isomorphic to � +n(g∗)⊗sn where ⊗s denotes +the symmetric tensor product. +2Occasionally we use the corresponding notation +∂ +∂ηi . +3We point out a possibly counter-intuitive quirk of our definition; for k ∈ △ the set Pk does not contain polyno- +mials of degree k but only those of degree less than k. +11 + +Lemma 2.9. For i, j ∈ {1,...,d}, +Xiη j = δi,j + +� +I̸=0, d(I)=sj −si +C I,ei +j +ηI , +(2.3) +where ei denotes the multi-index (0,...,0,1,0,...,0) with the 1 being in the i-th slot. +Proof. This follows directly from Proposition 2.5, applying Xi to the function +y �→ η j(xy) , +evaluating at y = 0 and using the fact that Xi|0 = +∂ +∂ηi |0. +Proposition 2.10 ([FS82, Prop. 1.26]). 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 . The analogous statement +holds for the vector fields Yj, +For a multi-index I = (i1,...,id) ∈ Nd we introduce the notation X I = X i1 +1 ...X id +d . Note that the +order of the composition matters since g is not in general Abelian. 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 . The next proposition follows as a direct consequence. +Proposition 2.11 ([FS82, Prop. 1.30]). The following maps from Pa → RdimPa are linear iso- +morphisms. +1. P �→ +�� +∂ +∂η +�I +P(e) +� +d(I) d(I) one has that +X I Pa +x[f ] = Pa−d(I) +x +[X I f ] +for all f ∈ C ∞(G). 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. +12 + +Theorem 2.13 (Taylor’s Theorem). 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). +Remark 2.14. Note that, while it is very useful to have a rather explicit form of the reminder in +order establish Schauder estimates in Section 3.5, there is a more common version of Taylor’s +Theorem on homogeneous groups, c.f. [FS82, Thm. 1.37], 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)| +(2.4) +and which follows as a trivial consequence of Theorem 2.13. Furthermore the analogous +claim for the right Taylor polynomial, (PR)a +x[f ], holds. In this case the analogue of (2.4) 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)| . +(2.5) +Remark 2.15. 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)| . +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)| +(2.6) +≲a,f +� +δ≥0 +λ−(a+δ)−|s||x−1y|a+δ , +where the sum over δ runs over some finite subset of (0,∞). +Remark 2.16. 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) . +Remark 2.17. For any p ∈ Pγ one has +˜Pγ +x[p](y) = p(y) . +13 + +Remark 2.18. In the proof of Theorem 2.13, given below, we use the following elementary +observations +• The map +Φ : Rn → G, +(t1,...,tn) �→ exp(t1X1)·...·exp(tnXn) +is a diffeomorphism. 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,...,tn) �→ +� +i +ti Xi. +Then since, +Φ(r s1t1,...,r sn tn) = r ·Φ(t1,...,tn) +it is seen to be a global diffeomorphism. +• 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 +(2.7) +uniformly over t ∈ Rd. +• 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 +(2.8) +and one has λi,J,I = 0 whenever d(J)+di ̸= d(I) . +• If a ∈ △, then ma := max{|I| : d(I) ≤ a} = [a] where [a] denotes the integer part of a. +This follows from the fact that d(I) ≤ a implies ma ≤ [a] and since ([a],0,...,0) is in the +set over which the maximum is taken. +Proof of Theorem 2.13. Define for i ∈ {1,...,d} the following measure on Rd with support on +Bs +|t|s(0) +˜Qei (t,ds) = +� +ji +δ0(ds j) +and define Qei (y, ·) = Φ∗ ˜Qei (Φ−1(y), ·) to be the push-forward measure. First we show that, +f (xy)− f (x) = +n� +i=1 +� +G +(Xi f )(xz)Qei (y,dz) . +(2.9) +14 + +Indeed one can write for y = y1y2... yd, where yi = exp(ti Xi) and t = (t1,...,td) = Φ−1(y) +f (xy)− f (x) = +d� +i=1 +f (xy1... yi−1yi)− f (xy1...yi−1) += +d� +i=1 +�ti +0 +∂s f (xy1... yi−1exp(sXi))|s=s′ds′ += +d� +i=1 +�ti +0 +(Xi f )(xy1 ... 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) . +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 (2.7), 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. We turn to the +proof in the general case. +Claim 2.19. 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. 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) . +We shall prove by induction on m ∈ {1,...,⌈a⌉}, that for multi-indices J satisfying a − m < +d(J) ≤ a the claim holds. +• The case m = 1 follows from (2.8) and (2.9), +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. +15 + +• Suppose the claim holds for m −1. 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 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 < +∞ , +(2.13) +where | · |Pα denotes any norm on the finite dimensional vector space Pα. +Furthermore,C(G) denotes the space of continuous functions (note that only the strict inclusion +C(G) ⊊ C0(G) holds). +Remark 2.21. We note that as in [Hai14], for α ∈ △ the Cα(G) spaces do not agree with the +usual notion of continuously α-differentiable functions. This definition, however, is more +canonical in the context of regularity structures, c.f. Theorem 3.12. +The following proposition gives some useful properties that also mirror those of Euclidean +Hölder spaces. +Proposition 2.22. Given any real number α ∈ R the following holds +1. Cα(G) ⊂ Cβ(G) for every β < α. +2. For any multiindex I the map X I : Cα(G) → Cα−d(I)(G), +f �→ X I f is well defined. +3. If α > 0, then any distribution f ∈ Cα(G) agrees4 with a continuous function. +Note that this proposition in particular implies that C ∞(G) = ∩α>0Cα(G). +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). +18 + +Proof. Points 1 and 2 follow directly. For Point 3, denote by ˜Px the polynomial in Defini- +tion 2.4, 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. Thus we find that the distribution f agrees with the continuous function x �→ ˜Px(x), +concluding the proof. +Remark 2.23. Note that by Proposition 2.22 for α > 0 and any f ∈ Cα(G) we know that X I f +agrees with a continuous function whenever d(I) < α. Thus, in particular Definition 2.3 of +Taylor polynomials Pα +x [f ] extends to Cα(G) ⊃C ∞(G). +2.3 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. Furthermore, the quotient map π : G → S +is a smooth normal covering map , c.f. [Lee13, Thm. 21.29] and the canonical G right action +S ×G → S, +(Gx,x′) �→ Gxx′ +makes it a homogeneous space. We call G a lattice if S = G/G is a compact space. This is +equivalent to S carrying a finite G invariant measure, which we shall denote by d(Gx), c.f. +[Rag07, Thm. 2.1]. Throughout this article we assume that every homogeneous Lie group we +work with carries a lattice, the following theorem, [Rag07, Thm. 2.12], gives sufficient condi- +tions for this to be the case, which all our examples satisfy. +Theorem 2.24. Let G be a simply connected nilpotent Lie group and g its Lie algebra. Then, G +admits a lattice if and only if g admits a basis with respect to which the structure constants of +g are rational. +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. 2.14] for an example. +Denote by π∗ : C(S) →C(G) the pull-back under π. 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 2.10. In order +to check that this map is well defined, we use the following terminology. 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) . +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. 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 . +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. +2.4 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. These will be revisited in +Section 4.4 when we discuss the heat operator on the Heisenberg group and Kolmogorov +type operators. +2.4.1 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 +� +� +. +Remark 2.25. 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 +� +. +One sees that these definitions are equivalent after identifying Cn with R2n. +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). 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. We observe that for any (x, y,z) ∈ Hn and +i = 1,...,n, +[Ai,Bi](x, y,z) = −2C. +(2.14) +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. +20 + +Definition 2.5. A homogeneous Lie group G is called stratified (or a Carnot group) if its Lie +algebra is graded and generated by W1. +Due to (2.14), we see that if we equip Hn with the dilation map, +λ·(x, y,z) := (λx,λy,λ2z), +then Hn becomes a stratified group. Refer to [Bra14, Sec. 3.3.6] for a discussion of generalisa- +tions of this structure. +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. Note that it is not a normal +subgroup and thus the quotient space Hn/Hn not a group. However, since it is a lattice the +homogeneous space Hn/Hn is compact, c.f. Subsection 2.3. +2.4.2 MATRIX EXPONENTIAL GROUPS +For n ≥ 1, let B be a rational n ×n block matrix of the form, +B = + + +0 +B1 +0 +··· +0 +0 +0 +B2 +··· +0 +... +... +... +... +... +0 +0 +··· +0 +Bk +0 +0 +··· +0 +0 + + +with each Bi a pi−1×pi block matrix of rank pi, where n ≥ p0 ≥ p1 ≥ ··· ≥ pk and �k +i=0 pi = n. +Note that this implies the zero blocks on the diagonal are all pi × pi square matrices. 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 +� +. +To define the dilation we begin by decomposing according to the structure of the block matrix +B, writing +Rn = Rp0 ×···×Rpn . +Thus the action of B⊤ on z = (z0,...,zk) ∈ Rp0×···×Rpk is written B⊤z = (B⊤ +1 z0, B⊤ +2 z1,...B⊤ +k zk−1) +etc. Then we set +λ·(t,z0,...,zk) := (λ2t,λz0,...,λ2k+1zk). +The origin e = (0,0) is again the identity element and (t,z)−1 = (−t,−exp(−tB⊤)z). The Lie +algebra is spanned by the translation invariant vector fields, +Xi(t,z) = ∂zi +for i = 1,...,p0 +and +Y (t,z) = ∂t −(Bz)·∇ = ∂t − +n� +i,j=p0+1 +bi j zi∂zi . +This defines the matrix exponential group associated to B and one sees that it is not stratified. +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). +The equivalent scaling as above is to set λ · (t,x,v) = (λ2t,λv,λ3x). We refer to [Man97] for +more details and Section 4.4 below for a discussion of natural, second order, hypoelliptic, +linear operators associated to these groups. +3 REGULARITY STRUCTURES AND MODELS +Definition 3.1. A regularity structure is a pair T = (T,G) consisting of the following elements: +1. 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. 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. +The space T is called the structure space. +2. 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 . +The group G is called the structure group of T . +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}}. +We make use of the natural shorthands, T>α, T≥α, T<α, T≤α and the projections Q>α etc. +Definition 3.2. 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;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;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 +. +22 + +Lastly, we denote by MT the space of models for T equipped with the semi-norms ∥M∥γ;K := +∥Π∥γ;K +∥Γ∥γ;K. +Remark 3.1. In the rest of the article, often without any further comment, we will write +r := min{n ∈ N : n > |min A|} . +3.1 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. 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. For a multi-index I, we write +ηηηI :=ηηηi1 +1 ·...·ηηηid +d as well as 1 :=ηηη0. 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 . 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 . +These maps are extended multiplicatively to all of ¯T. Using Prop. 2.5 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). +3.1.1 DERIVATIVES AND ABSTRACT POLYNOMIALS +Next we lift the vector fields X i to abstract differential operators X i on ¯T of degree sj, c.f. +Definition 3.6 given later. 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. The two con- +ditions to be an abstract differential operator are checked directly: +23 + +1. Indeed X i : Tα �→ Tα−si. +2. By the Leibinz rule the second property is checked by showing that +X iΓx,yηηηj = Γx,yX iηηηj . +(3.1) +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 (2.3) +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) . +Thus (3.1) follows from the injectivity of the maps Πx. +Remark 3.2. 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 3.6 in Section 3.4. +3.1.2 ABSTRACT TAYLOR EXPANSIONS +The following operator, which sends a smooth function to its abstract Taylor expansion, will +be used throughout the article. +Definition 3.3. 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. +Remark 3.3. Note that the mapPPPa +x is well defined by Remark 2.23 . We shall almost exclusively +use it, when its argument is a smooth function f ∈C ∞(G) ⊂ Ca(G). +Lemma 3.4. Let f ∈C ∞(G) and a ≥ 0. 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). Furthermore, for +b ≥ a one has +PPPa +x[f ] = Q≤aPPPb +x[f ] . +(3.2) +Proof. We first observe that (3.2) holds since the polynomial ΠeQ≤aPPPb +x[f ] satisfies the prop- +erty required in Definition 2.3. The first part of the lemma follows by combining Proposi- +tion 2.11 with (3.2). +Lemma 3.5. In the setting of the above lemma one has +|X IΠePPPa +x[f ](z)| = |X I Pa +x[f ](z)| ≲ +� +d(I)≤d(J) 0 and P ∈ ¯T<δ. 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. +Proof. We observe that Pδ +x[ΠeP] = ΠeP. Thus, the claim follows directly from Lemma 3.4. +3.2 MODELLED DISTRIBUTIONS +Definition 3.4. 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 +< +∞ . +We define the corresponding semi-norm ∥ · ∥γ;K on Dγ +M by setting, +∥f ∥γ;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 +. +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 ; ¯f ∥γ;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 +. +(3.3) +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;M. We will freely drop the explicit +dependence on the model, image sector and its regularity when the context is clear. +3.2.1 RECONSTRUCTION +Theorem 3.7 (Reconstruction Theorem). Let T = (T,G) be a regularity structure with α = +min A. 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;B2λ(x)∥Π∥γ;B2λ(x) , +(3.4) +25 + +uniformly over x ∈ K. 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 ; ¯f ∥γ;B2λ(x)∥ ¯Π∥γ;B2λ(x) ++∥f ∥γ;B2λ(x)∥Π− ¯Π∥γ;B2λ(x) +� +, +(3.5) +Remark 3.8. Existence of a reconstruction operator for γ ≤ 0 also holds, however, uniqueness +does not. In the case γ = 0 the analogous bound to (3.4) contains an additional logarithmic +correction on the right hand side, c.f. [CZ20]. +Remark 3.9. 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. +In order to prove Theorem 3.7 we require two preliminary results. As in [FH20, Sec. 13.4] +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, +(3.6) +where the δ here denotes the Kronecker delta applied componentwise to the multi-index. 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). We then have the following result. +Lemma 3.10. 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. +Proof. 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.f. (2.5) with a = 0) that +|f ∗ρ(m)(x)− f (x)| = +���� +� +G +(f (y)− f (x))ρ(m)(y−1x)dy +���� ≲ max +i=1,...,d ∥Yi f ∥L∞∥ρ∥L1r−m . +Secondly, since +Y Iρ(n,m) = Y I(ρ(n) ∗ρ(n+1,m)) = (Y Iρ(n))∗ρ(n+1,m) +(3.7) +one finds by applying Young’s convolution inequality m-times, c.f. Section 2.2, 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,...,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. 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 → +∞. Using (3.7) we obtain convergence +in D(G). It remains to check the support of ϕ(n). 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 . +It follows from the definitions that ϕ(n) = ρ(n) ∗ϕ(n+1). We set +˜ρ(m,n) := ˜ρ(m) ∗ ˜ρ(m−1) ∗···∗ ˜ρ(n) +and using (2.12) we note that ˜ϕ(m+1) ∗ ˜ρ(m) = ˜ϕ(m) and ˜ρ(m,n) → ˜ϕ(n) in D(G) as m → ∞. +Lemma 3.11. Let r > 1 and ρ be as in Lemma 3.10 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). Then the sequence ξn is Cauchy in C−β(G) for every β > α and +the limit ξ satisfies ξn = ξ∗ ˜ϕ(n). 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]. +Proof. The proof follows along the same steps as that of [FH20, Lem. 13.24], but one has to +be careful since convolution is non-commutative in our setting. Let λ ∈ (0,1] and ψ ∈ Br , we +first establish the bound +|〈ξn −ξn+1,ψλ +x〉| ≲ λ−βr(α−β)n +(3.8) +uniformly over ψ ∈ Br , λ ∈ (0,1] and locally uniformly over x ∈ G. First observe the trivial +bound, +|〈ξn −ξn+1,ψλ +x〉| ≤ sup +x∈K +(|ξn(x)|+|ξn+1(x)|)∥ψλ +x∥L1 ≤ (1+rα)CK ¯Crαn, +where ¯C := sup{ +� +|ψλ(x)|dx : ψ ∈ Br } < +∞ . Hence, when λ ≤ r−n the bound (3.8) holds +directly. +In the case r−n ≤ λ, using (2.11) we rewrite +|〈ξn −ξn+1,ψλ +x〉| = 〈ξn+1 ∗ ˜ρ(n) −ξn+1,ψλ +x〉| = |〈ξn+1,ψλ +x ∗ρ(n) −ψλ +x〉|. +By Taylor’s theorem and in particular Remark 2.15 +|ψλ +x(z)− ˜Pr +y[ψλ +x](z)| ≲r +� +δ>0 +λ−(r+δ)−|s||y−1z|r+δ , +(3.9) +where the sum runs over a finite set. It follows from (2.10), Remark 2.8 and (3.6) 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 (3.9) 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. 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. +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). +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. +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. Since r + δ > α the quantity on the left is +summable over k ≥ n and is of order λγ, concluding the proof. +We are now ready to proof Theorem 3.7. +Proof of Theorem 3.7. 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 +〉. +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) . +Using the identity +(F ∗ ˜φ)(x) = 〈F,φx〉 = +� +F(y)φx(y)d y , +it follows that +�R(m,n) f −R(m+1,n) f +� +(x) = +��R(m,m) f −R(m+1,m+1) f ∗ ˜ρ(m)� +(y)ρ(m−1,n) +x +(y)d y += +�� +Πy f (y)∗ ˜ϕ(m) −R(m+1,m+1)f ∗ ˜ρ(m)� +(y)ρ(m−1,n) +x +(y)d y += +��� +Πy f (y)∗ ˜ϕ(m+1) −R(m+1,m+1) f +� +∗ ˜ρ(m)� +(y)ρ(m−1,n) +x +(y)d y += +��� +Πy f (y)∗ ˜ϕ(m+1) −R(m+1,m+1) f +� +(z)ρ(m) +y +(z)dzρ(m−1,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,n) +x +(y)d y += +�� +〈Πy f (y)−Πz f (z),ϕ(m+1) +z +〉ρ(m) +y +(z)dzρ(m−1,n) +x +(y)d y . +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. +Then successively applying the facts that, +• supy,z∈Br−m (0) |〈Πyτ,ϕ(m+1) +z +| ≲ ∥Π∥γ;Br−m(0)r−αm|τ|ζ for τ ∈ Tζ, +• ∥f (y)−Γyz f (z)∥α ≲ ∥f ∥γ;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; ¯Kr−γm , +uniformly over m ≥ n ≥ 0, where ¯K denotes the two fattening of the set K. 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; ¯Kr−αn, +(3.10) +where α = min A. It follows that R(m,n) f converges uniformly on compacts as m → ∞ to +R(n) f which also satisfies the bound(3.10). 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) . +Therefore we may apply Lemma 3.11 to see that there exists a limit Rf := limn→∞ R(n) f . +With validity of the limit established we now turn to show the bounds (3.4) and (3.5); this +requires us to keep more careful track of the underlying sets in the proof. We begin with (3.4), +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 +(3.4) can be written as the claim that for all λ ∈ (0,1], +sup +ψ∈Bm +|〈R(f − fx),ψλ +x〉| ≲K λγ∥f ∥γ;B2λ(x)∥Π∥γ;B2λ(x) . +(3.11) +Using that |(f − fx)(z)|α ≲ ∥f ∥γ;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;Bλ(y)∥f ∥γ;BCr−n (y) +� +α≤α≤γ +r−αn|y−1x|γ−α +≲ ∥Π∥γ;Bλ(y)∥f ∥γ;BCr−n (y)r−αn(|y−1x| +γ−α +r(α−γ)n), +(3.12) +where C(r) := +r +r−1 is as in Lemma 3.10. Given n > n0(λ) sufficiently larger, we have a uni- +form bound Cr−n ≤ λ. 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 3.11, in par- +ticular noticing that we integrate the above estimate over y ∈ Bλ(x), we conclude that (3.11) +holds for the limit R. +Using the obvious notation we can also rewrite (3.5) as +sup +ψ∈Bm +|〈R(f − fx)− ¯R( ¯f − ¯fx),ψλ +x〉| ≲ λγ � +∥f ; ¯f ∥γ;B2λ(x)∥ ¯Π∥γ;B2λ(x) +∥f ∥γ;B2λ(x)∥Π− ¯Π∥γ;B2λ(x) +� +, +uniformly over λ ∈ (0,1]. This is seen very similarly to (3.11) 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;Bλ(y)∥f ; ¯f ∥γ;Bλ(y) +∥Π− ¯Π∥γ;Bλ(y)∥ ¯f ∥γ;Bλ(y) +� +� +α≤α≤γ +r−αn|y−1x| +γ−α . +It remains to show that the reconstruction map is unique for γ > 0 and that Rf is indeed an +element of C α. This is done exactly as in [Hai14, Sec. 3]. +30 + +3.2.2 FUNCTIONS AS MODELLED DISTRIBUTIONS +Let ¯T = (¯T, ¯G) be the polynomial regularity structure equipped with the polynomial model. +We show that in this setting the reconstruction theorem and its inverse map Taylor polyno- +mials to Hölder functions and vice versa. +Theorem 3.12. For γ > 0 the reconstruction operator is an isomorphism between Dγ(G) and +Cγ(G). In particular the inverse of the reconstruction map is given by +Cγ(G) ∋ f (·) �→PPPγ +(·)[f ] ∈ Dγ(G), +where PPPa is defined in Definition 3.3 +Proof. 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.e. Equation(3.4), and the definition of +Cγ in (2.13) that the reconstruction operator is a map Dγ(G) → Cγ(G). Continuity also follows +from (3.4) and by linearity. +To see the other direction recall that given a Hölder continuous distribution f ∈ Cγ(G) by +Proposition 2.22 the {X I f }d(I)<γ are actually functions and, in particular, that ˜Px = ΠxPPP[γ] +x [f ]. +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 +(3.13) +uniformly in ψ ∈ Br . 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) . +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) +����� . +(3.14) +Together, (3.13) and (3.14) imply that that |cI +x,y| ≲ λγ−d(I), we may then apply Lemma 3.6 to +see that PPP[γ] +(·)[f ] ∈ Dγ . +3.2.3 LOCAL RECONSTRUCTION +We will require the following further refinement of the reconstruction theorem which is an +analogue in our case of [Hai14, Prop. 7.2]. +31 + +Proposition 3.13. In the setting of Theorem 3.7 one has the improved bound, +sup +ψ∈Bm +|(Rf −Πx f (x))(ψλ +x)| ≲ λγ∥Π∥γ;B2λ(x) +sup +y,z∈supp(ψλ +x) +sup +ℓ<γ +|f (z)−Γzy f (y)|ℓ +|y−1z|γ−ℓ +, +(3.15) +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 ; ¯f ∥γ;supp(ψλ +x)∥ ¯Π∥γ;B2λ(x) ++∥f ∥γ;supp(ψλ +x)∥Π− ¯Π∥γ;B2λ(x) +� +, +(3.16) +for any x ∈ G and any λ ∈ (0,1]. +Proof. Since the right hand side of (3.15) is linear in f , as in [Hai14] we may assume it to be +equal to 1. We use the functions ρ(n) and ϕ(n) from Section 3.2.1 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. 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 3.10, 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. +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 . +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 +(3.17) +32 + +as well as for n > n0 +Rn−1 −Rn = +� +Λn−1 +Ππn−1(y) f (πn−1(y))(ψλ +xϕn−1 +y +)d y − +� +Λn +Ππn(z) f (πn(z))(ψλ +x ϕn +z )dz += +� +Λn−1 +Ππn−1(y) f (πn−1(y)) +� +ψλ +x +� +Br−n+1 +ρ(n−1)(w)ϕ(n) +ywdw +� +d y +− +� +Λn +Ππn(z) f (πn(z))(ψλ +xϕn +z )dz += +� +Br−n+1 +ρ(n−1)(w) +� +Λn−1 +Ππn−1(y) f (πn−1(y))(ψλ +xϕ(n) +yw)d ydw +− +� +Λn +Ππn(z) f (πn(z))(ψλ +xϕn +z )dz += +� +Br−n+1 +ρ(n−1)(w) +� +Λn−1 +Ππn−1(zw−1) f (πn−1(zw−1))(ψλ +xϕ(n) +z )dzdw +− +� +Λn +Ππn(z) f (πn(z))(ψλ +xϕn +z )dz += +� +Br−n+1 +ρ(n−1)(w) +� +Λn +� +Ππn−1(zw−1) f (πn−1(zw−1))−Ππn(z) f (πn(z)) +� +(ψλ +xϕn +z )dzdw . +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. +A similar argument gives |Rn| → 0 as n → ∞ which combined with (3.17) concludes the proof +of (3.15). The proof of the analogous bound, (3.16), follows in a similar manner. +3.3 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. Our main example will be modelled distributions on space-time domains that are al- +lowed to be discontinuous at {t = 0}, see Section 4. 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. +We fix a homogeneous sub-Lie group P ⊂ G with associated Lie algebra for which we write +p ⊂ g. 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. 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 ) . +(3.18) +Furthermore, we set +|x|P := 1∧dG(x,P) = 1∧inf +� +z ∈ P : |x−1z| +� +, +|x, y|P := |x|P ∧|y|P. +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. Given K ⊂ G we define the set +KP := +� +(x, y) ∈ (K\P)2 : x ̸= y and |x−1y| ≤ |x, y|P +� +. +That is KP contains all the points in K that are closer to each other than they are to P. +Definition 3.5 (Singular Modelled Distributions). 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;K := sup +x∈K\P +sup +ζ<γ +|f (x)|ζ +|x|(η−ζ)∧0 +P +, +�f �γ,η;K := sup +x∈K\P +sup +ζ<γ +|f (x)|ζ +|x|η−ζ +P +. +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;K := ∥f ∥γ,η;K + +sup +(x,y)∈KP +sup +ζ<γ +|f (x)−Γx,y f (y)|ζ +|y−1x|γ−ζ|x, y|η−γ +P +< +∞. +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 ; ¯f +������ +γ,η;K := ∥f − ¯f ∥γ,η;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 +. +If V is a sector of regularity α ∈ A, where appropriate we will use the shorthand Dγ,η +α;P,M = +Dγ,η +P,M(V ) and we will drop the dependence on the model when the context is clear. +Remark 3.14. We refer to [Hai14] for more intuition regarding the definition of these spaces +and their properties - all of which carry over to our setting. In particular [Hai14, Rem. 6.4] +discusses the relationship between the spaces Dγ,η +P +and Dγ. +Remark 3.15. The family of norms �f �γ,η;K and the two following lemmas play a role when we +consider fixed-point maps in Section 4 below. Their utility is in allowing us to extract small +scaling parameters in terms of the distance to the subgroup. In the semi-linear evolution +equation setting this allows us to obtain fixed points on sufficiently short time intervals. +34 + +Lemma3.16. 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]. 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. Then one has the bound, +�f �γ,η;K ≲ +������f +������ +γ,η;K, +where the implied constant depends affinely on ∥Γ∥γ;K but is otherwise independent of K. 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. Then one has the bound, +�f − ¯f �γ,η;K ≲ +������f ; ¯f +������ +γ,η;K +∥Γ− ¯Γ∥γ;K +�������f +������ +γ,η;K + +������ ¯f +������ +γ,η;K +� +, +with proportionality constant also depending affinely on ∥Γ∥γ;K and ∥¯Γ∥γ;K. +Proof. The proof follows almost exactly as that of [Hai14, Lem. 6.5]. For completeness we +provide a sketched proof of the first inequality, the second follows analogously. +Firstly, note that for x ∈ G such that dG(x,P) ≥ 1 or for ζ ≥ η both bounds follow directly +from the definitions. Hence we restrict our attention to x ∈ K such that dG(x,P) < 1 and +ζ < η. We define a recursive sequence by setting x0 := x, x∞ := ¯x = argminz∈P |x−1z| and +xn := x∞ +� +2−n ·(x−1 +∞ x0) +� +. 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| . +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. +(3.19) +The main difference in the proof is to apply (3.19) in place of [Hai14, Equation (6.4)], then the +rest of the proof adapts closely. One uses (3.19) together with the definition of +������f +������ +γ,η;K, to +show that for any ζ ∈ A, +|f (xn+1)−Γxn+1xn f (xn)|ζ ≲G +������f +������ +γ,η;K2n(η−ζ)|x|η−ζ +P +. +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. Applying the triangle inequality, the inductive hypothesis, the properties +of Γ and (3.19) we find, as in the proof of [Hai14, Lem. 6.5], that +|f (xn+1)− f (xn)|ζ ≲ 2n(η−ζ)|x|η−ζ +P +, +where the constant depends affinely on ∥Γ∥γ;K. 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). +Using that A is locally finite we may complete the induction which finishes the proof of the +first inequality. The proof of the second follows as in [Hai14, Lem. 6.5]. +35 + +Lemma 3.17. Let γ > 0, κ ∈ (0,1) and assume f , ¯f satisfy the assumptions of Lemma 3.16. +Then, for every compact K ⊂ G, one has +������f ; ¯f +������ +(1−κ)γ,η;K ≲ �f − ¯f �κ +γ,η;K +�������f +������ +γ,η;K + +������ ¯f +������ +γ,η;K +�1−κ +. +Proof. Follows by a direct adaptation of the proof of [Hai14, 6.6]. One need only replace Rd +there by G here. +3.3.1 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.e. 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). However, we show below that under appropriate assumptions +there exists a natural extension of ˜Rf to an actual distribution on G with regularity Cα. +Proposition 3.18 (Singular Reconstruction). Let f ∈ Dγ,η +P (V ), γ > 0 and η ≤ γ. Then, provided +α ∧ η > −m, where m is the scaled dimension of the complement of the singular hyperplane +defined by (3.18), 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. 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 ; ¯f +������ +γ,η; ¯K + +������M; ¯M +������ +γ; ¯K, +where the constant depends on semi-norms of f , ¯f and Z, ¯Z on ¯K. +We provide the proof of Proposition 3.18 at the end of this section, let us first make some +preparatory observations. Recalling the decomposition g = p ⊕pc defined at the start of the +subsection, we define the projections πc : g → pc and πp : g → p. 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). +(3.20) +Similarly to Remark 2.18 one sees that ˜Φ is a global diffeomorphism. We then define the map +NP : G → R+, +x �→ +��exp +� +πc ◦ ˜Φ−1(x) +��� +(3.21) +and observe the following properties. +• For x ∈ G one has that NP(x) = 0 if and only if x ∈ P . This follows from the fact that +P = exp(p). +• For x ∈ G and δ > 0 one has the identity +NP(δ· x) = δNP(x) . +(3.22) +Indeed, writing x = ˜Φ( ˜X ), we have +NP(δ· x) = |exp(πc(Dδ ˜X ))| = |exp(Dδ πc( ˜X ))| = |δ·exp(πc( ˜X ))| = δNP(x) . +36 + +• For any x ∈ G and y ∈ P one has +NP(yx) = NP(x) . +(3.23) +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 (2.1). Identity (3.23) then +follows from the fact that H(Y , ˜X p) ∈ p. +• The map NP is Lipschitz continuous on G and it follows from Remark 2.4 that NP is +smooth on G\P. +• There exists a constant C > 0 such that for all x ∈ G +CNP(x) ≤ dG(x,P) . +(3.24) +In a neighbourhood of the origin this follows directly from the fact that NP is Lipschitz +continuous. Homogeneity of Np (given by (3.22) above) and of the distance function +x �→ dG(x,P) then shows that this constant is in fact uniform on all of G. +• For all x ∈ G +dG(x,P) ≤ NP(x) . +(3.25) +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) . +Proof of Proposition 3.18. The proof is analogous to that of [Hai14, Lem.6.9], the only ele- +ment that does not adapt ad verbatim, is the construction of the partition of unity ϕx,n. We +therefore present an alternative construction of a partition of unity on G, which satisfies all +the required conditions. +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. +Secondly, let Z ⊂ P be a lattice (see Section 2.3) and let ˜ϕ be smooth, compactly supported, +such that +� +y∈Z +˜ϕy(x) = 1 +(3.26) +for all x in the 2 +C -fattening of P ⊂ G, where C is the constant in (3.24)and (3.25). For y ∈ Z we +then set +φy(x) := ϕ(CNP(x)) ˜ϕy(x) , +where the constant C is as in (3.24). To conclude, we then set for every n ≥ 0, y ∈ Z and x ∈ G, +φn,y(x) := φy(2n · x). +37 + +By using (3.22) and (3.23), it directly follows that φn,y(x) = (φ1,e) +� +y−1(2n · x) +� +. 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. 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 (3.26) in the last equality. The remainder of the proof of [Hai14, Lem. 6.9] then +adapts ad verbatim by also also making use of Proposition 3.13. +Remark 3.19. If the model M is smooth, i.e. Πxτ ∈ C ∞(G) for every τ ∈ T, one finds exactly as +in [Hai14, Rem. 3.15] 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). +3.4 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. These extensions adapt easily from [Hai14], thus for brevity we +provide them directly for singular modelled distributions. +3.4.1 DIFFERENTIATION OF SINGULAR MODELLED DISTRIBUTIONS +Definition 3.6. 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. +We say ∂ realizes L for the model M = (Π,Γ) if +• for any a ∈ V and x ∈ G, one has Πx∂a = LΠxa. +The following proposition is an immediate consequence of the definitions and the unique- +ness of the singular reconstruction operator. +Proposition 3.20. In the setting of Definition 3.6 let f ∈ Dγ,η +P (V ) for some γ > η, then ∂f ∈ +Dγ−β,η−β +P +. 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 . +Proof. For the first claim one may directly adapt the proofs of [Hai14, Prop. 5.28 & Prop. 6.15]. +The second claim follows by applying the Proposition 3.18 to Dγ−β,η−β +P +, +38 + +Remark 3.21. 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. 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. +Remark 3.22. Recall that in Section 3.1.1 we lifted the differential operators Xi crucially using +the additional structure of the polynomial regularity structure. This is by no means the only +lift, though certainly the most canonical one. 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. +3.4.2 MULTIPLICATION AND COMPOSITION WITH SMOOTH FUNCTIONS +We recall the notion of a product on a regularity structure. +Definition 3.7. 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 . +We say that a sector V is stable under the product if V ⋆V ⊆ V . 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. +We show that the product of two singular modelled distributions is again a singular mod- +elled distribution with possibly new regularity and singularity parameters. +Proposition 3.23. 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). Then f = f1 ⋆γ f2 ∈ Dγ,η +α;P with +α = α1 +α2, +γ = (γ1 +α2)∧(γ2 +α1), +η = (η1 +α1)∧(η2 +α1)∧(η1 +η2). +Here ⋆γ is the projection of ⋆ onto T<γ. +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 ;g +������ +γ,η;K ≲ +������f1;g1 +������ +γ1,η1;K + +������f2;g2 +������ +γ2,η2;K +∥Γ− ¯Γ∥γ1+γ2;K, +uniformly over any compact set K ⊂ G. +Proof. One may directly adapt the proofs of [Hai14, Th. 4.7] and [Hai14, Prop. 6.12], recalling +that the homogeneous distance satisfies a triangle inequality with a constant, c.f. Remark2.4. +39 + +We define the composition of modelled distributions with smooth functions as in [Hai14]. +Given a ∈ V , a function-like sector, we decompose a = ¯a1 + ˜a with ˜a ∈ T>0 and ¯a = 〈1,a〉. +Further, let ζ > 0 be the smallest non-zero value such that Vζ ̸= 0 so that we actually have +˜a ∈ T≥ζ. +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! +˜a⋆γk, +(3.27) +where we used the isomorphism T0 ∼ R for each component of a = (a1,...,an) ∈ V n, as well as +the notation ˜ai = ai −Q0ai. 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. +We make the same observation as in [Hai14, Sec. 4.2], that although the sum in (3.27) looks +infinite, since ζ > 0 we have ˜a⋆γk ∈ T≥|k|ζ and so only finitely many terms of the sum are +non-zero. In the following proposition we naturally extend the definition of a modelled dis- +tribution and associated norms componentwise. +Proposition 3.24. Let P be as above, γ > 0 and f = (f1,..., fn) ∈ Dγ,η +P (V n) be a collection of +modelled distributions for some function-like sector V which is stable under the product ⋆. Let +furthermore F ∈ C∞(Rn;R), then, provided η ∈ [0,γ], the modelled distribution +ˆFγ(f )(x) : G → V, +with ˆF(f ) defined as in (3.27), belongs to Dγ,η +P (V ). Furthermore, the map ˆFγ : Dγ,η +P (V ) → +Dγ,η +P (V ) is locally Lipschitz continuous in any of the semi-norms ∥ · ∥γ,η;K and |||·|||γ,η;K. +Furthermore, the analogous Lipschitz bound holds when working with two models. +Proof. One may follow ad verbatim the proof of [Hai14, Prop. 6.13], as well as [HP15, Prop. 3.11] +for the last sentence, since all Taylor expansions are carried out in Euclidean space. +3.5 CONVOLUTION WITH SINGULAR KERNELS +In this section we describe how to lift the action of singular kernels onto the regularity struc- +ture. 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. This latter issue is circumvented using Lemma 3.6 which allows us to reduce our +analysis to similar expressions as appear in [Hai14]. The examples we have in mind are the +singular part of Greens functions of left invariant differential operators satisfying the follow- +ing assumption. +Assumption 3.25. The kernel K : G\{e} → R can be decomposed as +K (x) = +� +n∈N +Kn(x) +(3.28) +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. +Remark 3.26. 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. 5.1 & Ass. 5.4] +adapted to our setting. Although Assumption 3.25 is somewhat less general we choose to +work with it for two reasons; firstly it is simpler to verify and secondly it highlights the role that +translation invariance plays in our applications. Lemma 4.4 which is an amalgam of [Hai14, +Lem. 5.5 & Lem. 7.7] 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 3.25 and a sufficiently well-behaved remainder. +Remark 3.27. Although we work explicitly with the one-parameter kernels of Assumption +3.25 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). Note that under this convention, for any +left-translation invariant vector field X , one has (X K )(f ) = X (K f ). +From now on we shall exclusively work with regularity structures T models M satisfying +the following assumption. +Assumption 3.28. 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. +We point out that the assumption K annihilates polynomials causes no real restriction on +the type of kernels since the result [Hai14, Lem. 5.5] adapts in a straightforward manner to +our setting, see also Lemma 4.4 below. We point out that this assumption is convenient but +not crucial for the theory, c.f. [HSa]. 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. Given a γ ∈ R∩△ we write Kγ for this lift; it corresponds to K in the sense that for +f ∈ Dγ +RKγ f = K (Rf ) +(3.29) +and in that it satisfies an appropriate version of the classical Schauder estimates. +41 + +Definition 3.8. 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. For each α ∈ A, I : Vα → Tα+β, where Tα+β := {0} for α+β ∉ A. +2. It annihilates polynomials, that is I : ¯T∩V → {0}. +3. For each τ ∈ T, Γ ∈ G : (I Γ−ΓI)(τ) ∈ ¯T. +Assume that the kernel K satisfies Assumption 3.25 for some β > 0. 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), +(3.30) +where the last sum is seen to converge absolutely by first observing that by Assumption 3.25 +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 3.4. +Definition 3.9. 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. +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. +Remark 3.29. Given a kernel K satisfying Assumption 3.25 a regularity structure and model +satisfying Assumption 3.28, 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 . This is the con- +tent of the extension theorem found as [Hai14, Thm. 5.14], which holds in our setting as well. +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 3.31, 3.32 and 3.33, so that the +interested reader will easily be able to fill in the remaining details. +The next theorem which is an analogue of [Hai14, Thm. 5.12], confirms that Kγ does indeed +correspond to K in the sense of (3.29) and satisfies the desired Schauder estimates. +42 + +Theorem 3.30. Let γ ∈ R \△ and β > 0 be such that γ + β ̸∈ △, let K : G \{e} → R be a kernel +satisfying Assumption 3.25 for r ≥ γ+β, let T = (T,G) be a regularity structure and M = (Π,Γ) +be a model satisfying Assumption 3.28. Furthermore assume that T is equipped with an ab- +stract integration operator and M realises K for I. 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 ). 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 ; ¯Kγ ¯f ∥γ+β;K ≲C +������f , ¯f +������ +γ; ¯K +∥Π− ¯Π∥γ; ¯K +∥Γ− ¯Γ∥γ; ¯K , +where the implicit constant depends on the norms of M, ¯M and f ∈ Dγ +M(V ) and ¯f ∈ Dγ +¯M(V ). +The fact that the next lemma ([Hai14, Lem. 5.16]) 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.f.[HSa]. +Lemma 3.31. In the setting of Theorem 3.30 one has the identity +Γx,y(I +J (y)) = (I +J (x))Γx,y +for all x, y ∈ G. +Proof. 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]. +We introduce the following quantity; 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) +(3.31) +where the second equality follows from Remark 2.12. Here we reiterate that we are using the +notation K (x, y) := K (y−1x) where K satisfies Assumption 3.25. Taylor’s theorem and Remark +2.15 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) . +(3.32) +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,·))− ˜Pα+β +x +[Kn(Πxτ)](y) +� +φ(y)d y += +� +n +�� +Πxτ(Kn(y,·))− ˜Pα+β +x +� +(Πxτ)2(Kn(·,·)) +� +(y) +� +φ(y)d y += +� +n +�� +Πxτ(Kn(y,·))−(Πxτ)(˜Pα+β +x,1 [Kn(·,·)](y) +� +φ(y)d y += +� +n +� +Πxτ +� +Kn(y,·)− ˜Pα+β +x,1 [Kn(·,·)](y) +� +φ(y)d y += +� +n +� +Πxτ(K 0,α +n,xy)φ(y)d y +where we use the subscript in (Πxτ)2(Kn(·,·)) to clarify that this denotes the function w �→ +Πxτ(Kn(w,·)) and the analogous subscript for ˜Px,1[K (·,·)](y) to clarify that one expands in the +first coordinate. The next Lemma collects the results of [Hai14, Lem. 5.18, Lemma 5.19], the +proofs of which adapt ad verbatim. +Lemma 3.32. 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) , +(3.33) +where the sum over δ runs over a finite set of positive real numbers. The same bound holds for +|(Πxτ)(K I,α +n,xy)|, as well as the analogue bound on the difference of two models. +Furthermore one has the bound +� +n +���� +� +G +(Πxτ)(K I,α +n,xy)φλ +x(y)dy +���� ≲ λα+β∥Π∥α;B2(x)(1+∥Γ∥α;B2(x)) , +(3.34) +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) +(3.35) +Lemma 3.33. Under the assumptions of Theorem 3.30 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). Again, the analogous bound holds for the difference +of two models holds as well. +Proof. 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. Observe that since J n +x,yτ ∈ ¯T<α+β it suffices by Lemma 3.6 +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| . +(3.36) +We treat the cases |y−1x| ≤ 2−n and |y−1x| > 2−n separately. 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) . +Using the analogous notation J n(x) for the operator consisting only of one summand in +(3.30), we find +ΠxJ n(x)Γx,yτ(z) = +� +γ≤α +Πx +�J n(x)QγΓx,yτ +� +(z) += +� +γ≤α +ΠxPγ+β +x +[ +� +ΠxQγΓx,yτ +� +2Kn(·,·)](z) += +� +γ≤α +˜Pγ+β +x +[ +� +ΠxQγΓx,yτ +� +2Kn(·,·)](z) += +� +γ≤α +ΠxQγΓx,yτ +� +˜Pγ+β +x,1 [Kn(·,·)](z) +� += (Πyτ) +� +˜Pα+β +x,1 [Kn(·,·)](z) +� +− +� +γ≤α +ΠxQγΓx,yτ +�˜Pα+β +x,1 [Kn(·,·)]− ˜Pγ+β +x,1 [Kn(·,·)] +� +(z) += ˜Pα+β +x +[(Πyτ)2Kn(·,·)](z)− +� +γ<α +ΠxQγΓx,yτ +�˜Pα+β +x,1 [Kn(·,·)]− ˜Pγ+β +x,1 [Kn(·,·)] +� +(z) +and +ΠxΓx,yJ n(y)τ(z) = ΠyJ n(y)τ(z) = ˜Pα+β +y +[(Πyτ)2Kn(·,·)](z) += ˜Pα+β +x +� +˜Pα+β +y +[(Πyτ)2Kn(·,·)] +� +(z) += ˜Pα+β +x +�� +Πyτ +� +2 +�˜Pα+β +y,1 [Kn(·,·)] +�� +(z) , +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. 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 3.32 we find that +� +n :|y−1x|≤2−n +|X I pn +τ,1(e)| ≲ |y−1x|α+β−d(I)|τ| . +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,·) +� +. +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 3.28 +we have γ+β ∉ △. This concludes the case |y−1x| > 2−n. +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)). +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 3.5 +|X I qn +2 (e)| ≲ +� +d(I) 2−n� +. +Proof of Theorem 3.30. 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 . +(3.37) +For a ∉ △ this bound follows from Lemma 3.33 and properties of modelled distributions by +an ad verbatim adaptation of the same argument in the proof of [Hai14, Thm. 5.12]. The only +46 + +place, where the proof [Hai14, Thm. 5.12] does not adapt directly is when showing the bound +for a ∈ △. As in the proof of Lemma 3.33 we use Lemma 3.6 to circumvent the difficulty of +not having explicit Taylor expansions in our setting. The rest of the proof adapts almost ad +verbatim. +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 +(3.38) +and thus, in order to prove (3.37) for the polynomial part it suffices by Lemma 3.6 to show +that for any d(I) < γ+β, p(e) := ΠeP satisfies +|X I p(e)| ≲ |y−1x|γ+β−d(I) . +(3.39) +Using (3.28) we define the analogous decompositions J = � +n J n and Nγ = � +n N n +γ and simi- +larly write P = � +n=0 Pn, etc. +As usual we use different strategies for small and large scales, starting with the case 2−n ≤ +|y−1x|. In this case we separately estimate pi(e) := ΠePi for i ∈ {1,2,3} . +• 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 3.5 +|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) . +• 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 3.7, +|X I pn +2 (e)| ≲ 2−n(γ+β−d(I)) . +• 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)) . +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) . +Next we turn to the case 2−n > |y−1x|; here a computation analogous to the one preceding +[Hai14, Eq. (5.48)] 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,·) +� +(3.40) +which can be bounded exactly the way it is done there. +Finally, one may follow the concluding steps of the proof of [Hai14, Thm. 5.12], 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 ). +The proof of the difference bound follows by similar steps as to those presented here and +applied in the proof of [Hai14]. +3.6 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 3.3 on singular modelled distributions. +Proposition 3.34. In the setting of Section 3.3 and Theorem 3.30, 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|. +Furthermore, if γ+β ∉ △ and η+β ∉ △, one has Kγ f ∈ Dγ+β,(η∧α)+β +P +continuity bound +������Kγ f ; ¯Kγ ¯f +������ +γ+β,(η∧α)+β;K ≲ +������f ; ¯f +������ +γ,η; ¯K +∥Π− ¯Π∥γ; ¯K +∥Γ− ¯Γ∥γ; ¯K +(3.41) +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. +Proof. The proof follows exactly along the same lines as the one of [Hai14, Prop. 6.16], where +as in the proof of Theorem 3.30 the only modification needed is due to the fact that our Taylor +expansions are non explicit. Using the same notation as in the proof of Theorem 3.30, we +recall (3.38) +P := Γx,yNγf (y) +� +�� +� +=:P1 +−Nγ f (x) +� �� � +=:P2 ++J (x)(Γx,y f (y)− f (x)) +� +�� +� +=:P3 +. +(3.42) +This time, in order to show that Kγ f ∈ Dγ+β,(η∧α)+β +P +, by Lemma 3.6, 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 +, +(3.43) +There are three scales, which need separate arguments. +48 + +• In the case 2−n ≤ |y−1x| one proceeds exactly as in the proof of Theorem 3.30 using the +decomposition pn = �3 +i=1 pn +i . +• In the cases 2−n ∈ +� +|y−1x|, 1 +2|x, y|P +� +and 2−n ≥ 1 +2|x, y|P one uses Equation (3.40) and +proceeds exactly as in [Hai14] from there onwards. +Again, the proof of the difference bound follows analogously. +3.7 SYMMETRIES +As in [Hai14, Sec. 3.6], we shall consider modelled distributions which respect certain sym- +metries of G. 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 2.3 acting by G×G ∋ (n,x) �→ (nx) ∈ G. We +extend this to an action on function ψ : G → R by pull-back, i.e. +(n∗ψ)(x) := ψ(n−1x) = ψn(x). +For a regularity structure T = (T,G) we give the following definition, by analogy with [Hai14, +Def. 3.33] but restricted to this more straightforward setting. +Definition 3.10. 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. +A modelled distribution f : G → T is said to be symmetric if f (nx) = f (x) for every x ∈ G and +n ∈ G. +The following proposition is an amalgam of [Hai14, Prop. 3.38 & Prop. 5.23]. +Proposition 3.35. 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 3.10). Then, for every mod- +elled distribution f ∈ Dγ, for some γ > 0, symmetric with respect to the action of G, the follow- +ing hold; +1. For every φ ∈C ∞ +c (G) and n ∈ G one has (Rf )(n∗φ) = (Rf )(φ). +2. For any x ∈ G and n ∈ G one has (Kγ f )(nx) = (Kγ f )(x). +Proof. The proof of the first point follows exactly the steps of the proof of [Hai14, Prop. 3.38] +in our simplified setting. +The proof of the second follows similarly along the lines of the proof of [Hai14, Prop. 5.23], +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. +49 + +3.8 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. The +proof of the next proposition, [Hai14, Prop 3.31], adapts ad verbatim to our setting +Proposition 3.36. Let T = (T,G) be a regularitystructure and (Π,Γ) a model. 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. 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; ¯K∥Γ∥α; ¯K, +as well as the analogous difference bound. +Furthermore, one has the following simple consequence of Lemma 3.31 and Lemma 3.33. +Proposition 3.37. Under the assumptions of Theorem 3.30 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; ¯K)∥Π∥α; ¯K + +sup +x,y∈K:|y−1x|<1 +|Γx,yτ|δ +|τ|α−β|y−1x|α−δ . +Again, the analogous bound for the difference of two models holds as well. +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 4.4 below for illustrative examples. 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. In line with this decomposition, in this +section we deviate from the convention stated above Equation (2.2) and reorder the basis of g +so that we always have the time component, i.e. X1 ∈ pc, in the first slot. Under this assump- +tion we also fix the basis ¯X = {Xi}d +i=2 of p where sXi = si . We note that P := exp(p) ⊂ G is a +homogeneous subgroup as in 3.3 and we recall the notation defined there, +¯s := s|p +and +|¯s| := trace(s|p). +In particular we have |s| = s1 +|¯s|. From now on we identify the Lie group G with R×P under +the diffeomorphism +R×P → G, +(t,x) �→ x exp(t X1) . +and we shall often use the notation z = (t,x) ∈ R×P = G. Let us collect some useful facts; +• Under this identification we see that R×{e} �� G is a homogeneous subgroup. +• 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. +50 + +• The map ˜Φ from (3.20) 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)). +• Recall the map NP : G → R+ defined by (3.21). There exists a constant c′ > 0 such that +NP(t,x) = c′|t| +1 +s1 . In particular, by (3.24) and (3.25) 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 . +(4.1) +A core assumption for the remainder of this section will be that of non-anticipativity. +Assumption 4.1. 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. +We will also require an assumption of prescribed homogeneity on the kernel, with respect +to the dilation map. +Assumption 4.2. 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). +We now have the following analogue of [Hai14, Lem. 7.4]. +Lemma 4.3. Given G : G\{e} → R satisfying Assumption 4.1 and Assumption 4.2 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 +� +. +(4.2) +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. +(4.3) +Proof. The proof follows exactly the same steps of [Hai14, Lem. 7.4] after replacing the usual +derivatives there with the vector fields ¯X = {Xi}d +i=2. +Lemma 4.4. Let G : G\{e} → R satisfy Assumption 4.1 and Assumption 4.2 with σ = −|¯s|. Then, +for any r > 0, there exist smooth functions K : G \ {e} → R and K−1 : G → R, both satisfying +Assumption 4.1 and such that G = K +K−1 and +• K is compactly and satisfies Assumption 3.25 with β = s1. +• K−1 : G → R is globally smooth and such that X IK−1 ∈ L∞ +loc(R;L1(P)) for any I ∈ Nd. +Proof. The proof readily adapts from [Hai14, Lem. 5.5 & Lem. 7.7] with the only real change +being to skip the last step in the proof of [Hai14, Lem. 7.7] and instead using Faa di Bruno’s +formula along with (4.3) to obtain the claimed integrabillity of K−1. +Remark 4.5. Note that more careful analysis yields improved bounds on K−1, however, K−1 ∈ +L∞ +loc(R;L1(P)) suffices for our purposes. +51 + +4.1 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. Note that using (4.1) there exists a constant c > 0 +such that [0,cT ]×P ⊂ OT . Hence, the corresponding notion of local solutions, described in +Section 4.3, corresponds to the usual one. +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. From now on we shall also +use subspaces of the previously introduced Hölder spaces (Definition 2.4), modelled distri- +butions (Definition 3.4) and singular modelled distributions (Definition 3.5) 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). We shall often write the corresponding +semi-norms for example as +∥ · ∥α;OT ∩K, +|||·|||γ;OT ∩K, +|||·|||γ;η;OT∩K , +where in particular we allow K = OT . +With these notions at hand the proof of the next theorem adapts directly from the proof of +[Hai14, Thm. 7.1]. +Theorem 4.6. 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 4.1) such that the assumptions of +Theorem 3.30 are satisfied for some β > 0 and a sector V of regularity α > −s1. Then, for every +T ∈ (0,1], and η > −s1 and for κ > 0 small enough +������KγR+ f +������ +γ+β,(η∧α)+β−κ;OT ≲ T κ/s1������f +������ +γ,η;OT +(4.4) +������KγR+ f ; ¯KγR+ f +������ +γ+β,(η∧α)+β−κ;OT ≲ T κ/s1 +�������f ; ¯f +������ +γ,η;OT + +������M; ¯M +������ +γ;O2 +� +. +(4.5) +The constant in the first bound depends only on |||M|||γ;O2, while in the second it may also de- +pend on +������f +������ +γ,η;OT ∨ +������ ¯f +������ +γ,η;OT and |||M|||γ;O2 ∨ +������ ¯M +������ +γ;O2. +Proof. The proof follows almost ad verbatim the proof of [Hai14, Thm. 7.1], only using our +modified definition of the sets OT := {z ∈ G : dG(z,P) ≤ T }. +Remark 4.7. In fact, given any closed set K ⊂ G the bounds (4.4) and (4.5) both hold if OT is +replaced on the left hand side by OT ∩K and on the right hand side by OT ∩ ¯K. However, we +will not make use of this fact in this article. +While Theorem 4.6 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. +52 + +Lemma 4.8. 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. Then, under +the assumptions of Theorem 4.6 one has the bound +��� +��� +���PPPγ +(·)[K−1RMR+ f ] +��� +��� +��� +γ+β,¯η;OT ≲ T +������f +������ +γ,η;OT +as well as the analogous difference bound +��� +��� +���PPPγ +(·)[K−1RMR+ f ];PPPγ +(·)[K−1R ¯MR+ ¯f ] +��� +��� +��� +γ+β,¯η;OT ≲ T +�������f ; ¯f +������ +γ,η;OT + +������M; ¯M +������ +γ,O1 +� +. +Proof. The argument adapts mutatis mutandis form the proof of [Hai14, Lem. 7.3], replacing +the compact support assumption therein by the integrability property of K−1. +4.2 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. 7.2]. +Lemma 4.9. 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 4.1 and 4.2. 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. Then, for every γ > 0 the map +PPPγ +(·)[G(u0)] : G\P → T belongs to Dγ,α +0;P for every γ > (α∧0) and furthermore, for any T > 0, one +has +��� +��� +���PPPγ +(·)[G(u0)] +��� +��� +��� +γ,η;OT < ∞. +Proof. We first decompose the kernel as in Lemma 4.4 and write +G(u0)(t,x) = K (u0)(t,x)+K−1(u0)(t,x) . +For the summands of K (u0)(t,x), one can argue as in the proof of [Hai14, Lem. 7.5] or as in +Section 3.5. The desired bound for K−1(u0)(t,x) follows exactly as in Lemma 4.8. +4.3 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 7.8] as well as the other parts of [Hai14, Section 7.3] +adapt to the setting of homogeneous Lie Groups. 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. This result covers Ander- +son type equations, such as the one treated in Section 6. Recall here the notation introduced +at the start of this section, in particular the identity |s| = s1 +|¯s|. +53 + +Theorem 4.10. Let α ∈ (−s1,0], γ > −α, η + α > −s1 and G : G \ {e} → R be a kernel satisfy- +ing Assumptions 4.1 and 4.2 with σ = −|¯s| and the decomposition G = K + K−1 as in Lemma +4.4. 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. Then, given a model M = (Π,Γ) which realises K for I and +satisfies the assumptions of Theorem 3.30 along with any v ∈ Dγ,η +0;P,, the map +MT ;v : Dγ,η +0 +→ Dγ,η +0 +U �→ KγR+(UΞ)+Pγ +(·)[K−1R+(UΞ)]+ v, +(4.6) +is well-defined and there exists a unique fixed point for some T > 0. +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 3.7), then the unique fixed point is as well. +Crucially, the solution map depends continuously on the model. +Remark 4.11. Note that using that the abstract mild equation given by (4.6) is linear in U, the +local time of existence T > 0 can be chosen independ of the initial condition. 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. We refer to +[HL18, Theorem 5.2] for details. +Remark 4.12. In practice we will usually take v = Pγ[G(u0)] so that by Lemma 4.9 the con- +dition v ∈ Dγ,η +0;P is satisfied provided u0 ∈ Cη(P), where η + α > −s1. 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 4.6 and K−1 the smooth +remainder, see Lemma 4.8. +Proof sketch of Theorem 4.10. The proof is a straightforward adaptation of the proof of [Hai14, +Theorem 7.8] and follows by applying Proposition 3.23, Theorem 4.6 and Lemma 4.8 to show +that the map M ¯T ;v has a unique fixed point in Dγ,η +0 +for some T > 0. Since α = min A, the +modelled distribution x �→ Ξ is an element of Dγ,γ +α;P for any γ > 0. Assuming U ∈ Dγ,η +0;P by +Proposition 3.23 one therefore has, +UΞ ∈ Dγ+α,η+α +α;P +. +So by using the singular Schauder estimate, Proposition 3.34, along with the assumption (η+ +α)∧α > −s1, one has +K(UΞ) ∈ Dγ+α+β,((η+α)∧α)+β +(α+β)∧0;P +. +Then, we see that +γ+α+β > γ, +η < η+α+β +and +(α+β)∧0 > 0 . +Hence, one finds Dγ+α+β,((η+α)∧α)+β +(α+β)∧0;P +�→ Dγ,η +0;P which concludes the proof that MT ;v is a self- +mapping on Dγ,η +α;P. By comparing MT ;vU and MT ;v ¯U for U, ¯U ∈ Dγ,η +0 +, one then shows that +MT ;v is a contraction for sufficiently small T > 0. +From the steps outlined above one furthermore obtains the the claims of G-periodicity and +the continuous dependence of the solution on the model. +54 + +4.4 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 1.1. In this +section we present a non-exhaustive list of homogeneous Lie groups and linear differential +operators, L, to which our results apply. Generically, these equations are of the form, +Lu = ∂tu − ¯Lu = F(u, ¯Xu,...,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 1.1, +c.f. [Fol75]. +Setting 1 P is a stratified Lie Group (see Definition 2.5) with a basis {Xi }m +i=1 of W1, generating the +Lie Algebra. 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). The heat type operator associated to the +sub-Laplacian +L = ∂t − +m +� +i=1 +X 2 +i . +satisfies the assumptions of Theorem 1.1. and it follows from [Fol75, Thm. 3.1 & Prop. 3.3] +that L is non-anticipative. We shall discuss a notable example of the setting, that of the +heat operator on the Heisenberg group, below. +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. A natural class of operators is given by +LQ = ∂t −Q(X1,..., 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. 7.8]. A notable example is the heat +type operator associated to the bi-sub-Laplacian on a stratified Group. +Setting 3 The homogeneous Lie Group G = R×P is equipped with a non-trivial Lie group struc- +ture. In this case we look at differential operators of the form, +L = X0 −Q(X1,..., Xm) . +where X0 = ∂t + ¯X0 and the operators L, L∗ satisfy the criteria of Folland’s theorem +and L is non-anticipative. Examples are parabolic/restricted Hörmander operators, +c.f. [Hai16, Def. 1.1] or [Bel95, Ch. 3] and the kinetic Fokker–Planck operator fits into +this setting. +55 + +4.4.1 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 2.4, is a stratified Lie group. 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. 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. 2.1] there exists a unique, fundamental solution K : Hn → R associated +to L which is smooth away from e and satisfies the assumptions of Lemma 4.4. As a result our +framework allows for the study of semi-linear evolution equations of the form, +∂tu −Lu = F(u, A1u,..., Anu,B1u,...,Bnu,ξ), +u|t=0 = u0. +4.4.2 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. The group structure is a matrix exponential group, as defined in Section +2.4. Given two, rational, n ×n block matrices, +A = + + +A0 +··· +0 +... +... +... +0 +··· +0 + +, +B = + + +0 +B1 +0 +··· +0 +0 +0 +B2 +··· +0 +... +... +... +... +... +0 +0 +··· +0 +Bk +0 +0 +··· +0 +0 + + +. +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. 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. Equipping R×Rn with the matrix exponential group struc- +ture associated to B (and defined in Section 2.4) it is readily checked that L satisfies the as- +sumptions of Theorem 1.1. In fact, there is an explicit formula for the fundamental solution. +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 +� +. +56 + +The kinetic Fokker–Plank operator falls into this class. 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 +� +. +Then the associated operator is +Lu(t,v,x)= ∂tu(t,v,x)−∆vu(t,v,x)− v ·∇xu(t,v,x). +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 +� +. +See [IK64, Sec. 7] for a derivation in the case d = 1 and [Man97] in the general case. +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 , +(5.1) +as treated by Theorem 4.10, where I denotes the abstract part of the lift of a 2 regularising +kernel as in Section 3.5, Ξ is the lift of a noise of regularity α and U0 is the polynomial part of +U. 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. 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) +(5.2) +one can work with essentially a truncated version algebraic framework as on the abelian +group Rd developed in [BHZ19]. Therefore, in the remainder of this short section, we freely +use notations and definitions from [BHZ19], often without further explanation. +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,...,sn), where n = max{i ≤ d : si < 2}. 5 Motivated by (5.1) we +define the naive (normal) rule, c.f. [BHZ19, Def. 5.7] +˚R(t) = {(t,Ξ),(t),(Ξ),()}, +˚R(Ξ) = {()} +(5.3) +and consider its completion R constructed in [BHZ19, Prop. 5.21] (note that this second +step is only necessary if α ≤ −3/2). We denote by T ex and T ex ++ ,T− ⊂ T ex +− +the corresponding +spaces constructed in [BHZ19, Def. 5.26, Def. 5.29, Def. 6.22] and summarise some important +properties of these spaces. +5Recall form Section 2 that s1, s2,..,sn are the first n eigenvalues of the scaling s in increasing order and that Xi +are the corresponding eigenvectors +57 + +Proposition 5.1. The triple {T ex,T ex ++ ,T−} have the following algebraic structure. +• The spaces T ex ++ ,T−,T ex +− +are graded Hopf algebras with coproduct given by △+ and △− +respectively. +• The space T ex is a right comodule over T ex ++ . +• The spaces T ex and T ex ++ +are left comodules over T− . +• These spaces satisfy the co-interaction property, i.e. 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. 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. 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. To circumvent these considerations we restrict ourselves to working +with the following subspaces. +Let T consists of the span of those basis vectors (trees) T ∈ T ex where T and its grading |T |+, +c.f. [BHZ19, Def. 5.3], 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. +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. +Observe that Proposition 5.1 still holds with T ex ++ +replaced by T+ and T ex replaced by T. We +define G+ to be the character group of T+. From now on our regularity structure shall be given +by +T = (T,G) , +(5.4) +where G consists of the elements Γ ∈ L(T,T ) of the form Γ = (id⊗ f )△+ for some f ∈ G+. +58 + +5.1 SMOOTH MODELS +In order to define the models, we have to deviate slightly from [BHZ19]. For i ∈ {1,...,n} we +write ηηηi instead of Xi for abstract polynomials and we declare a map ΠΠΠ : T → C ∞(G) (c.f. +[BHZ19, Def. 6.8]) to be admissible for a smooth function ξ ∈ C ∞(G) and a kernel K as in +Assumption 3.25, if for every x ∈ G +ΠΠΠΞ(x) = ξ(x), +Πηηηi(x) = ηi(x) +and +ΠΠΠIτ = KΠΠΠτ, +ΠΠΠIiτ = (XiK )ΠΠΠτ , +where we write I resp. Ii for the operation of attaching an edge of type t, resp. (t,ei) to the +root. 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,...,n}, c.f. [BHZ19, Lem. 6.9]. 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.e is reduced in the sense of [BHZ19, Def. 6.2.1]. 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) . +(5.5) +The renormalisation group G− is defined to be the character group of the Hopf algebra T ex +− . +We observe that in our case T ex +− +coincides (as an algebra) with the free algebra spanned by +a family of linear trees. (In the case α > −3/2, one furthermore has T ex +− = T− and we list the +trees explicitly in the next section). The group G− acts on models as follows. For g ∈ G− let +Rg = (g ⊗id)△−, we set +ˆM g = ( ˆΠg, ˆΓg) = (ΠRg,ΓγRg ) . +(5.6) +One can check that every element the orbit of a smooth canonical model as in (5.5) is indeed +a model. +Remark 5.2. Note that in order to show convergence of a family of models ˆM(ε) = ( ˆΠ(ε), ˆΓ(ε)) +of the form (5.6) 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 3.36 and 3.37. +We emphasise again that this section relied on the the assumption γ < 2. +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. Let G be a stratified Lie group (recall Definition 2.5) with Lie algebra g and +59 + +G a lattice subgroup with its canonical left action on G. Recall from Section 2.3 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.6 For m ≤ d, let X1,..., 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. Finally we +recall the notion of convolution ∗S on S induced by the right action of G on S, as defined in +Section (2.3). +We shall solve equations where the driving noise satisfies the following assumption for +some ¯α > 0. +Assumption 6.1. 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. +Remark 6.2. For ξ satisfying Assumption 6.1, 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 . +In particular, coloured noise where the regularisation is given by a negative fractional power +of the sub-Laplacian fits into our setting, c.f. [BOTW22]. +Remark 6.3. 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]. 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]. A similar remark holds for the choice of +mollifier ρ in Theorem 6.4 below. +In the above setting, we have the following theorem. +Theorem 6.4. Fix T > 0 arbitrary and let ξ be a noise satisfying Assumption 6.1 with ¯α ∈ +(−3/2,−1) and η ∈ (−(2 + ¯α),0). 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) , +(6.1) +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. 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. +6This is for example achieved by replacing the homogeneous norm on G with a multiple of itself. We only require +this condition in order to make the integrands in Section 6.1 supported on the fundamental domain K . +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. +60 + +Remark 6.5. We point out that the proof of Theorem 6.4 given below modifies mutatis mu- +tandis to the case where the noise is allowed to depend on time. In this case the natural mod- +ification of Assumption 6.1 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. The only dif- +ference in the proof is to replace all convolutions and integrals over S with convolutions and +integrals over R×S. 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 6.7 below. +Remark 6.6. The range of ¯α considered in Theorem 6.4 does not cover the full subcritical +regime of the Anderson equation on a stratified Lie group. Indeed one expects an analogue of +Theorem 6.4 to hold for any ¯α ∈ (−2,0). 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. 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 6.1. +Remark 6.7. 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. This probabilis- +tic notion of solution circumvents the need for explicit renormalisation as in (6.1). Further- +more, their results are obtained on the infinite volume group Hn, rather than the compact +quotient space that we consider here. We expect that by using weighted modelled distribu- +tions, c.f. [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 6.4. 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. +Proof of Theorem 6.4. We follow a usual strategy in the theory of regularity structures, show- +ing the equivalent theorem for the pulled back equations on G. For this, we work with the +regularity structure T = (T,G) constructed in (5.4) with |Ξ| = (−3/2, ¯α). Note that since we are +in Setting 1 we can directly apply Theorem 1.1 to see that there exists a unique, fundamental +solution G : (R×G)\(0,e) → R and satisfying the assumptions of Lemma 4.4, so that we have +G = K +K−1 enjoying the properties described therein. +Denote by M(ε) = (Π(ε),Γε) the canonical model for T , satisfying Π(ε) +z Ξ = ξε and realising +K for I. From now on we shall use the common tree notation, with the obvious changes in +interpretation, as defined for example in [HP15, Sec. 4.1], and write +:= Ξ, += IΞ, += ΞIΞ, +etc. +Since ¯α ∈ (−3/2,−1) the rule defined in (5.3) is complete and T− coincides (as an algebra) +with the free commutative algebra generated by +� +, +, +� +. 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 +� +, +� +. +61 + +For each ε ∈ (0,1) we fix gε ∈ ˜G− to be the character specified by +gε( +) = E[ξε(e)K (ξε)(e)] . +Thus, applying Theorem 4.10, we obtain a solution Uε : [0,T ]×G → T<γ to the abstract lifted +equation, (4.6), for the model ˆM(ε) := M gε +ε . Next we show that uR +ε := R ˆM(ε)Uε actually solves +Equation (6.1) in several steps. +• 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. In particular ˆΠ(ε) +z Uε(z)(z) = +u0 +ε(z). +• We observe that therefore +(Uε )(z) = u0 +ε(z) +� ++ ++ +� ++ +� +si<γ +ui +ε(z)ηηηi +. +• 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) . +• Thus, we can conclude using Remark 3.19 that +R ˆM(ε)(U g ) = u0 +ε(z)(ξε(z)−cε) = +�R ˆM(ε)U g � +(z)(ξε(z)−cε) . +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 6.8, stated below. +Proposition 6.8. The familly of models ˆM(ε) = ( ˆΠ(ε), ˆΓε) constructed in the proof of Theorem 6.4 +converges as ε → 0. +Proof. Note that by Remark 5.2 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 3.10 together with the estimates obtained +in Proposition 6.10. +Remark 6.9. For a detailed proof of a Kolmogorov type criterion for general models on Eu- +clidean domains, i.e. G = Rd, in the spirit above, we refer to [HSb, Appendix B]. +62 + +6.1 STOCHASTIC ESTIMATES FOR THE RENORMALISED MODEL +In this subsection we obtain convergence the renormalised models. We shall freely use tech- +niques of [Hai14, Sec 10] and as well as graphical notiation similar to that in [HP15, Sec. 5], +with some slight changes in interpretation. For example we write +� +Π(ε) +(0,e) +� +(ϕλ) = ++ +, +with the following interpretations; +• The node +represents the origin, (0,e) ∈ R×G while the edge +represents integra- +tion against the rescaled test function ϕλ. +• The node +represents an instance of the G left periodic white noise on G while the +edge +represents the kernel c ¯α ∗ρε. Note therefore, that when ε = 0 our diagrams +will still contain dotted lines, as opposed to [HP15]. +• 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 . 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. +As in [Hai14, Sec. 10.2] and [HP15, Sec. 5.1.1] we will also use the notation W(ε;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;k)τ for the renormalised model. For example, we find that ˆW(ε;2) +((s, y);x1,x2) = +W(ε;2) +((s, y);x1,x2) is given by +W(ε;2) +((s, y);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. +Proposition 6.10. Fix δ > 0 small enough. Then, for every λ ∈ (0,1), ϕ ∈ Br and z = (t,x) ∈ +R×G, there exist random variables +( ˆΠz )(ϕλ), +( ˆΠz +)(ϕλ), +( ˆΠz +)(ϕλ) +(6.2) +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) . +Proof. Firstly, we note that by translation invariance of the noise it is enough to consider +z = (0,e). Furthermore, since all random variables belong to a finite Wiener chaos, it suffices +63 + +to show these estimates for p = 2, [Hai14, Lem. 10.5]. The first two estimates, 1a) and 1b), +follow readily by Ito’s isometry. +For the next items we recall [Hai14, Lem. 10.14 & 10.18], the natural analogues of which +hold in our setting as well. 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) = +− +. +(6.3) +The first summand is the graphical representation of the iterated integral, I2(( ˆ +W(ε;2) +)(z)) +integrated against a test function centred at (0,e) ∈ R×G; see for example the analogous sec- +ond term in [Hai14, Eq. (10.23)]. The second summand is the graphical analogue of the first +term in [Hai14, Eq. (10.23)] given by I0(( ˆ +W(ε;0) +)(z)). Next note that by virtually an identical +calculation as in the beginning of the proof of [Hai14, Thm. 10.19] one finds, +|〈(( ˆ +W(ε;1) )(z),( ˆ +W(ε;1) )(¯z)〉| ≲ |z|2 ¯α+4 +|¯z|2 ¯α+4 , +(6.4) +which implies +|〈( ˆW(ε;2) +)(z),( ˆ +W(ε;2) +)(¯z)〉| ≲ (|z|2 ¯α+4 +|¯z|2 ¯α+4)|¯z−1z|2 ¯α . +Together with the simple estimate I0(( ˆ +W(ε;0) +)(z))| ≲ |z|2 ¯α+2 this gives 2a). +Next we turn to show 3a). First, note that by a similar calculation as for (6.4) one finds +|〈( ˆW(ε;2) +)(w),( ˆW(ε;2) +)( ¯w)〉| ≲ |w|2(4+2 ¯α) +| ¯w|2(4+2 ¯α) . +Therefore, together with the bound I0(( ˆ +W(ε;0) +)(w))| ≲ |w|2 ¯α+4 we find +E +��� +� +( ˆΠ(ε) +(0,e) +)⋄( ˆΠ(ε) +(0,e) ) +� +(ϕλ) +��� +2 ≲ λ2(4+3 ¯α) . +As in [HP15, Sec. 5.3.2] we find +� +ˆΠ(ε) +(0,e) +� +(ϕλ) = ++ + + +− + + ++ +− +and introduce the notationQε(s, y) = K (s, y)(cα∗ρε)∗2(y). We see that cε = +� +R×GQε(z)dz, and +define the renormalised kernel RQε(s, y) := Qε(s, y)−cεδ(0,e). Then, as in [HP15, Eq. (5.14)], +using the notation +for the renormalised kernel, one finds +� +ˆΠ(ε) +(0,e) +� +(ϕλ) = ++ +− ++ +− +. +64 + +Similarly, +� +( ˆΠ(ε) +(0,e) +)⋄( ˆΠ(ε) +(0,e) ) +� +(ϕλ) = +− +and thus it only remains the bound the covariance of the diagrams +, +, +, +(6.5) +The latter two can be bounded by repeatedly using the analogue of [Hai14, Lem. 10.14], while +for the first diagram we use additionally the natural analogue of [Hai14, Lem. 10.16]. This +concludes the proofs of 3a) and therefore the first column. The bounds on the random vari- +ables in (6.2) are obtained analogously by replacing c ¯α ∗ρε by just c ¯α throughout (and appro- +priately interpreting the renormalised kernel, Qε, for ε = 0). 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