The parabolic Anderson model on Riemann surfaces

We show well-posedness for the parabolic Anderson model on $2$-dimensional closed Riemannian manifolds. To this end we extend the notion of regularity structures to curved space, and explicitly construct the minimal structure required for this equation. A central ingredient is the appropriate re-interpretation of the polynomial model, which we build up to any order.


Introduction
The last few years have seen an explosion of literature on singular stochastic partial differential equations (singular SPDEs). The simplest instance of such an equation is the parabolic Anderson model in two dimensions, formally written as ∂ t u = △u + uξ. (PAM) Here u : [0, T ] × D → R is looked for, where D is some 2 dimensional domain, and ξ is (timeindependent) white noise on the domain D. This equation is formally ill-posed (or "singular"), since u is not expected to be regular enough for the product uξ to be well-defined analytically. The standard tool of stochastic calculus, the Itō integral, is also of no use here, since the whitenoise is constant in time.
With the breakthrough results of Hairer [Hai14] and Gubinelli, Imkeller and Perkowski [GIP15] a large class of such equations has become amenable to analysis. Let us sketch the approach of [Hai14], since this is the one we shall use in this work.
• assume that u "looks like" the solution ν to the additive-noise equation which is classically well-defined via convolution with the heat semigroup P t * University of Cambridge; the author is responsible for the first part of the Appendix † Max-Planck Institute Leipzig ‡ University of California San Diego; the author is responsible for the second part of the Appendix • under this assumption, if we somehow define ν · ξ, then the framework defines u · ξ automatically • close the fixpoint argument, i.e.
1. u "looks like" ν 2. w := P t u 0 + t 0 P t−s [u s ξ]ds 3. then w "looks like" ν It then only remains to define the missing ingredient "ν · ξ". This can be done probabilistically and is actually the only place in this theory that is not deterministic. Using this procedure, it is shown in [Hai14] that (PAM) possesses a unique solution for D = T 2 , the two dimensional torus.
In this work we show that the theory can be adapted to work for D = M , a 2-dimensional closed Riemannian manifold. The theory of regularity structures is intrinsically a local theory (as opposed to the theory of paracontrolled distributions, which, at least at first sight is global in spirit). It is hence natural to expect that it can be applied to general geometries. It turns out that the implementation of this heuristic is not straightforward.
At least two hurdles need indeed to be bypassed. On the one hand, at the core of Euclidean regularity structures stands the space of polynomials, encoding classical Taylor expansions at any point. The operation of re-expansion from a point to another leads to a morphism from (R d , +) to a space of unipotent matrices. On a manifold, one would need to look for such a space, encoding Taylor expansion and enjoying a similar structure. On the other hand, as usual for fixpoint arguments of (S)PDEs, one needs to estimate the improvement of the heat kernel in adequate spaces, which is a global operation (Schauder estimates).
To solve the first issue, we show that the space of polynomials on the tangent space of the manifolds is a suitable candidate for a canonical regularity structure, that allows to encode Hölder functions. This choice enforces a modified definition of a regularity structure. In particular one has to abandon the idea of one fixed vector space and work with vector bundles instead. For our definition of a model, there is no unipotent structure anymore and re-expansions are only approximately compatible. Within this new framework, when considering the parabolic Anderson model on a surface, we give a weak version of a Schauder estimate with elementary tools and heat kernel estimates.
This exposition does not demand any previous knowledge of regularity structures on the Reader. In this sense it is self-contained, apart from a reference to the reconstruction theorem of Hairer in our Theorem 20 and in the construction of the Gaussian model in Section 8. Its proof using wavelet analysis is of no use reproducing here. We believe that the validity of that reconstruction theorem, which we use in coordinates, is easily believed.
We follow a very hands-on approach. Instead of trying to set up a general theory of regularity structures on manifolds, we work with the smallest structure that is necessary to solve PAM. We show the Schauder estimates explicitly. Apart from introducing for the first time regularity structures on manifolds, we believe our work also has a pedagogical value. Since everyting is laid out explicitly and covers the flat case M = T 2 , it can serve as a gentle introduction to the general theory.
In future work we will investigate the algebraic foundation necessary for studying general equations, without having to build the regularity structure "by hand". For general equations a new proof of the Schauder estimates has also to be found.
During the writing of the present article, a different approached has been put forward in [IB2016a], where the notion of paracontrolled products using semi-groups is developed on general metric spaces. The advantage of the paracontrolled approach is that it requires less machinery. On the downside, the class of equations that can be covered is currently strictly smaller than in the setting of regularity structures. Let us point to [IB2016b] though, which pushes the framework to more general equations.
The outline of this paper is as follows. After presenting notational conventions, we give in Section 2 the notion of distributions on manifold we shall use in this work. Moreover we introduce Hölder spaces on manifolds. In Section 3 we introduce the notion of regularity structure, model and modelled distribution on a manifold. We show how these objects behave nicely under diffeomorphisms and use this fact to show the reconstruction theorem. In Section 4 we give the simplest non-trivial example of a regularity structure on a manifold; the regularity structure for "linear polynomials". This forms the basis for the regularity structure for PAM, which is constructed in Section 5. As input it takes the product νξ alluded to before. This is constructed in Section 8 via renormalization. Section 6 gives the Schauder estimate for modelled distributions in the setting of PAM and finally Section 7 solves the corresponding fixpoint equation. In Section 9 we show how the construction of Section 4 can be extended to "polynomials" of arbitrary order.

Notation
In all what follows M will be a d-dimensional closed Riemannian manifold. When we specialize to the parabolic Anderson model (PAM), the dimension will be d = 2. Denote by δ > 0 the radius of injectivity of M .
For τ ∈ G, G a graded normed vector bundle with grading A we denote by ||τ || a the size of component in the a-th level, a ∈ A.
The differential of a smooth enough function f : M → R at a point p will be denoted d| p f ∈ T * p M . Similar for higher order derivatives (see Section 9) ∇ ℓ | p f ∈ T * p M ⊗ℓ . For the action on vectors where B η (0) := {x ∈ R d : |x| < η}. Here r will be depend the situation, and will always be large enough so that the distributions under consideration can act on ϕ.
We shall use p, q for points in M and x, y, z to denote points in R d . For which is consistent with the notation introduced above when considering R d as Riemannian manifold with the standard metric.
For γ ∈ R we denote by ⌈γ⌉ the smallest integer strictly larger than γ.
For a pairing of a distribution T with a test function we write T, ϕ .
For two quantities f, g we write f g if there exists a constant C > 0 such that f ≤ Cg. To make explicit the dependence of C on a quantity h, we sometimes write f h g.
Proof. Let φ z , z ∈ Z d , be a partition of unity of R d such that supp φ z ⊂ B 1 (z) and sup z∈Z d ||φ z || C r < ∞.
Definition 10. Let M be a closed Riemannian manifold. Let a finite partition of unity (φ i ) i∈I be given on M , subordinate to a finite atlas (Ψ i , U i ) i∈I . For γ ∈ R, define For γ > 0, an equivalent characterization of C γ (M ) will be shown in Theorem 90. We now give one in the case γ ≤ 0.
Lemma 11. For γ ≤ 0, M a closed Riemannian manifold, an equivalent norm on C γ (M ) is given by where we recall that ϕ λ p is defined in (2).
Proof. Fix an atlas (Ψ i , U i ) with subordinate partition of unity φ i . Denote Indeed, this follows from The result then follows from Remark 7.
(C 1 ≤ C 2 ): We have to show ). Indeed, since I is finite and for all i ∈ I, φ i is compactly supported Away from the boundary, the differential of Ψ i is bounded, and then for z ∈ supp h i one has d(z, Ψ −1 i (x)) = O(λ). This proves the claim. Now one checks that h i;λ • exp Ψ −1 i (x) falls under Remark 7, and applies Remark 6.
As immediate consequence we get the following statement.
Corollary 12. Let (Ψ j ,Ū j ) j∈J be another finite atlas with subordinate partition of unity (φ j ) j∈J . Then for γ ≤ 0 with equivalent norms. We now give our definition of a regularity structure and a model on a manifold M . For concrete incarnations of these abstract definitions we refer the reader to Section 4 for the implementation of a first order "polynomial" structure; to Section 9 for a structure implementing "polynomials" of any order and right before Lemma 33 for the structure used for the parabolic Anderson model.

Regularity structures on manifolds
Definition 13 (Regularity structure). A regularity structure is a graded vector bundle G on M , with a finite grading A = A(G) ⊂ R. For α ∈ A, G α denotes the vector bundle of homogeneity α. It is assumed to be finite dimensional. We denote the fiber at p ∈ M by G| p and the fiber of homogeneity α at p by G α | p . For p ∈ M, τ ∈ G| p , α ∈ A we write proj Gα|p for the projection of τ onto G α | p .
Definition 14 (Model). Let a collection of open sets U q ⊂ M , q ∈ M , with q ∈ U q , and maps be given. We assume there is for every compactum K ⊂ M a constant δ K = δ K (Π, Γ, {U q } q ) > 0, such that Γ p←q is defined for p, q ∈ K, d(p, q) < δ K and for q ∈ K, exp q | B δ K (0) is a diffeomorphism and exp q (B δ K (0)) ⊂ U q .
Remark 15. Note that the conditions on a model do not pin down the global regularity of Π q τ . Without loss of generality we will assume that Π q τ ∈ C α (U q ) for all q ∈ M, τ ∈ G| q and α := min A(G).
Our definition of a regularity structure and a corresponding model are slightly more general than the original formulation by Hairer [Hai14]. This extension is necessary to accomodate the "polynomial regularity structure", which will be constructed up to first order in Section 4 and up to any order in Section 9. Let us point out the key differences.
• Derivatives of functions on a general manifold M can only be stored in a fibered space. Hence the regularity structure has to be a vector bundle and not a fixed vector space.
• For this reason there cannot be a fixed structure group G in which the transport maps Γ p←q take value.
• The transport maps Γ p←q can also act "upwards", see Remark 81.
• The distributions Π p τ as well as the transports Γ p←q only make sense locally.
It turns out that the theory can handle these slight extensions. In particular the reconstruction theorem still holds, Theorem 22. Finally, we remark that our regularity structure does not include time and that the parabolic Anderson model will be treated by considering functions in time, valued in modelled distributions (Definition 17) on a manifold.
As in Lemma 8 we know how Π p τ acts on a more general class of functions: Lemma 16. For a regularity structure G let be given a model (Π, Γ) of transport precision β with β ≥ sup a∈A(G) |a|. Let p ∈ K, a compactum in M . Let ϕ satisfy the assumptions of Lemma 8 with the additional condition supp ϕ ⊂ B δ K /4 (0) ⊂ R d . Assume moreover that B δ K /2 (p) ⊂ K (which can always be achieved by making δ K smaller.) Then for τ ∈ G ℓ | p Note that in the sum |λ K z| < δ K /2. Hence, by assumption q := exp p (λ K z) ∈ K. Hence by definition of a model, |Γ p←q τ | m ≤ ||Π, Γ|| β,K d(p, q) (ℓ−m)∨0 for τ ∈ G ℓ | q . Then for those z Definition 17. Let G be a regularity structure and (Π, Γ) a model of precision β ∈ R. Define for γ > sup α∈A(G) |α| the space of modelled distributions Here δ K is the distance of points in K for which Γ makes sense, see Definition 14. Note that the precision of transport β plays no role here.
Remark 18. As usual for Hölder norms, for every compactum K an equivalent norm is obtained by replacing in the supremum, for any δ ′ ∈ (0, δ K ], the condition d(p, q) < δ K with the condition d(p, q) < δ ′ .
Proof. Since Ψ has derivatives bounded below and above for every compactum, one can choose for every compactum K a constantδ K as in the definition of a model, such thatΓ p←q is welldefined for p, q ∈ K and d(p, q) < δ K as well as exp N q (B δ K (0)) ⊂Ū q . Here exp N denotes the exponential map on N .
1. Let q ∈ K ⊂ N and τ ∈G a | q and ϕ ∈ B r,δ K since ϕ λ q • Ψ • exp q falls under Remark 7. For p, q ∈ K ⊂ N with d(p, q) < δ K and τ ∈G| q , we have again by Remark 7. Finally for p, q ∈ K ⊂ N with d(p, q) < δ K and τ ∈G a | q , we have and similarily for the distance of two modelled distributions.
Lemma 20 (Reconstruction for M ⊂ R d ). Let G be a regularity struture on M , an open connected subset of R d . Let (Π, Γ) be a model with precision β ∈ R. Let γ > 0 and assume β ≥ γ. Denote α := inf A. Assume either that α < 0, or that α = 0 and that the lowest homogeneity in G is given by the constant distribution (of the polynomial regularity structure of Section 4).
For every f ∈ D γ (M, G) there exists a unique Rf ∈ C α (M ) such that for every compactum Here ϕ ∈ B r,δ K , r > |α|, (so that the action of Π x f (x) is well-defined) and K := B δ K (K).
Remark 21. Uniqueness actually holds in the class of operators R that satisfy (5) with γ replaced by any θ > 0.

Proof. Existence
We will apply [Hai14, Proposition 3.25]. 1 This Proposition is formulated for R d , but the statement is local and also holds for M ⊂ R d . So we have to verify for ζ uniformly over x, y ∈ K, n ≥ n 0 , n 0 = log 2 (δ K )∨0 and 2 −n ≤ |x−y| ≤ δ K . In [Hai14,Proposition 3.25] the upper bound 1 is chosen on |x − y|, but any upper bound works, so we chose δ K , since we need Γ x←y to be well-defined.
Here ϕ n x := 2 nd/2 ϕ(2 n (· − x)), and ϕ is a scaling function for a wavelet basis of regularity r > |α|. We have chosen n 0 also such that for n ≥ n 0 and x ∈ K, τ ∈ G| x the expression Π x τ, ϕ n x is well-defined. First, (7) follows from the fact that α is the lowest homogeneity in A(G) (note that ϕ n x is scaled to preserve the L 2 -norm, whereas the scaling in the definition of a model preserves the L 1 -norm).
We bound the first term as 1 Compare also [Hai15, Theorem 2.10] for a concise presentation of the (wavelet) techniques involved in its proof.
since 2 −n ≤ |x − y|. The second term is bounded as This proves (6) and an application of [Hai14, Proposition 3.25] gives the existence of Rf satisfying the bound (5).
The preceding argument is valid for α < 0. For α = 0, one can run the argument for some α ′ < 0 and get unique existence of Rf ∈ C α ′ with the claimed properties. In Corollary 23 below it is shown that actually Rf ∈ C 0 .
Lemma 22 (Reconstruction for M a closed Riemannian manifold). Let M be a closed Riemannian manifold with regularity structure G and (Π, Γ) a model with transport precision β ∈ R. Let γ > 0, and f ∈ D γ (M, G) and assume β ≥ γ.
Denote α := inf A. Assume either that α < 0 or that α = 0 and that the lowest homogeneity in G is given by the constant distribution (of the polynomial regularity structure).
Proof. By a cutting up procedure, it is enough to show (8) for ϕ ∈ B r,δ ′ , with δ ′ ∈ (0, δ M ] to be chosen. Let (Ψ i , U i ) i∈I an atlas with subordinate partition of unity (φ i ) i∈I , with I finite. On each chart, we push-forward the regularity structure, model and f to Ψ i (U i ), with corresponding reconstruction operationR i , modelΠ i and modelled distributionf i . For each i ∈ I, fix a compactum K i ⊂ U i such that supp φ i is strictly contained in K i . By Lemma 19, Now reconstruct in each coordinate chart asT i :=R ifi using Theorem 20. Define Rf := falls under Remark 7 around z. Summing over i gives (8).
Corollary 23. In setting of the previous theorem, assume that the lowest homogeneity in G is 0 and that it is given by the constant (as in the polynomial regularity structure of Section 4). Then Rf is given by projection onto that homogeneity, i.e.
Proof. DefineRf (p) := f 0 (p), then Recall that the projection proj is defined in Definition 13. The last term is of bounded by a constant times λ η , where η is the smallest homogeneity strictly larger than 0.
For the second to last term we first write By the properties of a model Hence |f 0 (·) − f 0 (p)| d(·, p) η∧γ and then Hence, by Remark 21,R = R.
We want to apply the Lemma 22 to the terms in the heat kernel asymptotics (Theorem 40). The problem is that their support will be of order 1 (and not of order λ as for ϕ λ x ). Hence we need the following refinement which is similar to Lemma 8.
Lemma 24. In the setting of Lemma 22, let ϕ satisfy the assumptions of Lemma 8 with the additional condition supp Proof. Let ϕ z,λ be given as in the proof of Lemma 16 with K : Note that in the sum |λ M z| < δ M /2. Hence exp p (λ M z) ∈ M is well-defined. Now the first summand can be written as The second summand is bounded as 4 Linear "polynomials" on a Riemannian manifold The regularity structure for linear "polynomials" on the Riemannian manifold M will be built on the vector bundle (M × R) ⊕ T * M . For readability introduce the symbol 1 and decree that it forms a basis for R. Define the graded vector bundle q M be the fiber at q. A generic element of T q will be written as Note that since R1 is a trivial fiber bundle, it is enough to specify it on the basis element 1. This is not possible on T * M . Note also that Π q ω is chosen to have value 0 and differential ω at q.
Finally define the re-expansion maps Γ p←q : T q → T p as , which is well-defined for d(p, q) < δ; δ the radius of injectivity of M . Π and Γ together form the polynomial model, where we take δ M = δ in Definition 14.
The transport of ω ∈ T * y M is chosen such that Π q ω and Π p Γ p←q ω have, at p, the same value and the same first derivative. Our re-expansion is not exact, i.e. we do not have Π q τ = Π p Γ p←q τ , but we have the following.
By construction f (p) = g(p), df (p) = dg(p) and hence the statement follows from Taylor's theorem.
Remark 26. In the setting of the previous Lemma, not only f (p) = g(p) but also f (q) = g(q). Indeed, for two points p, q ∈ M, at distance smaller than the cut locus and ω q ∈ T * y M , where the tangent map satisfies indeed d| p (exp −1 q ) : is the unique path from p to q, with length and speed d(p, q), staying within the cut-locus from y, that is (exp p (tv p )) 0≤t≤1 : in other words, for any 0 ≤ t ≤ 1, Hence, The next lemma follows from Lemma 25 and is shown in more generality in Theorem 89.
Lemma 27. The above is a model of transport precision β = 2.
As a sanity check for our construction, we mention the following lemma, which is almost immediate in the flat case (see [Hai14, Lemma 2.12]). We will prove it in Section 9 in a more general setting.

The regularity structure for PAM on a manifold
In the next four sections M is a 2-dimensional closed manifold.
The regularity structure for PAM will be built on two copies of the vector bundle, M × R 2 ⊕ T * M. We denote these two copies by V and W. In order to distinguish the different elements of these bundles we introduce the symbols {1, Ξ, I[Ξ], I[Ξ]Ξ} and decree that they form a basis for R 4 . We then write where ΞT * M is simply another copy of T * M. Formally we have, V = WΞ. As usual we will let T | p , V| p , and W| p denote the fibers of these bundles over p ∈ M.
The vector bundles V and W are graded, with gradings for some α ∈ (−3/2, −1) corresponding to the regularity of the driving white noise ξ.
is the projection taking an element to its β -component. To be concrete, generic elements τ ∈ V| p , τ ′ ∈ W| p are of the form with a, b, d, e ∈ R, c, f ∈ T * p M . And then for example All the graded fibers have a canonical norm, where on the cotangent space we use the norm induced by the Riemannian metric. For β ∈ A, τ ∈ V| p (or τ ∈ W| p ) we write, in a slight abuse of notation, |τ | β := | proj β τ |.
The model we shall use for the parabolic Anderson model will be time dependent, so we need slight extensions of our definitions.
Definition 29. For G = V, W, assume we are given a family of models where ||Π t , Γ t || β,M is defined in Definition 14. Note that for fixed t, the model comes with a reconstruction operator (Theorem 22), which we shall denote R t .
Definition 30 (Time-dependent modelled distributions). For G = V, W, given a family of models (Π t , Γ t ) parametrized by t ∈ [0, T ], denote by D t,γ (G) = D t,γ (M, G) the corresponding spaces of modelled distributions. That is, as defined in Definition 17, For N > 0, define the modified norm Here Remark 31. The modified norms with scaling parameter N are necessary for the fixpoint argument, see Remark 36.
As usual with Hölder-type spaces on compact domains, these spaces are complete Banach spaces.
We now build the model for the structures V, W. As input we need realization of Ξ and I[Ξ]Ξ.
Definition 32. Assume for T > 0 we are given ξ ∈ C α (M ) and a family of distributions where the action of the heat kernel p on ξ is well-defined by Theorem 34. Define where r := ⌈|α|⌉ and δ is the radius of injectivity of M .
In our application to white-noise forcing, ξ will be the white noise on M and Z will be constructed via Gaussian renormalization in Section 8. Now define the models for V and W as Lemma 33. These are in fact models with δ M = δ the radius of injectivity of M and the distances/norms of the model only depend on ξ, Z. Indeed for G = V, W, γ ∈ R ||Π t,G || β;γ;M 1 + ||ξ, Z|| α,2α+2,T , with β = 2 for G = W and β = 2 + α for G = V.
Analogously, one gets the bounds for W with β = 2.

Schauder estimates
Let p be the heat kernel on M . We start with a Schauder estimate for distributions. Since its proof follows the same idea as the upcoming Schauder estimate for modelled distributions, we omit the proof of the next theorem.
Theorem 34. Let T > 0, and F ∈ L ∞ ([0, T ], C α (M )), for α ∈ (−2, −1). Then for t ∈ [0, T ] We now prove an extension of this classical result to the space of modelled distributions. For The well-definedness of these terms is part of the following theorem.
2 Recall from the beginning of this section that fα, f2+2α are real-valued and f1 α is a section of T * M .
Remark 37. Contrary to classical Schauder estimates, we only get an "improvement of 4/3 derivatives". In order to get an "improvement of 2 derivatives" one has to include quadratic polynomials in the regularity structure. This is also the reason why we have to choose γ, γ 0 in such a specific way. Note that an improvement by 4/3 will be enough to set up the fix-point argument.
To be specific, in order to get an "improvement of 2 derivatives" the complete list of symbols necessary is, ordered by homogeneity, where i, j = 2, 3 stand for the space-directions. 3 These symbols would be the building blocks for the regularity structure on flat space. On a manifold the polynomials would represent the respective symmetric covariant tensor bundles, as laid out in Section 4. The Schauder estimate has to be shown on the level of each of theses symbols, and hence a treatment "by hand" as we do here would be cumbersome.
Remark 38. The following proof based on the heat kernel (almost) being a scaled test function goes back, in the flat case, to [CM2016]. A proof splitting up the heat kernel into a sum of smooth, compactly supported kernels (following the strategy of [Hai14]) is also possible, but more cumbersome.
Proof of Theorem 35. The first statement follows from the definition of h 0 and the fact that reconstruction of modelled distributions taking values only in positive homogeneities is given by the projection onto homogeneity 0, see Lemma 23.
Recall that δ M = δ, the radius of injectivity. By Remark 18 we can, and will only consider d(p, y) < δ/4. Introduce the short notation We shall need the following facts. Since where we used the classical Schauder estimate Theorem 34.
Moreover for a function ϕ satisfying the assumptions of Lemma 16 and Lemma 24 (recall that R t is the reconstruction operator of Theorem 22 associated to the model (Π t , Γ t )) and similarily We now estimate each term in the definition of the norm ||Kf || Dγ ,γ 0 ,N T (W) . Space regularity where p = p N + R N using heat asymptotics, Theorem 40.

Regarding the easier term involving
where ∇ acts on the dummy variable • and convolution acts on · and γ is the geodesic connection q to p. Since this expression is well-defined for N large enough and of order We now treat the term involving p N . Denoting by g(t, s) the integrand of the above integral, for The first term we bound as where we used (10) together with Lemma 41 (i), as well as to Hölder continuity of f α in space (9) and in time.
The second we bound as where we used (10) together with Lemma 41 (i) as well as the Hölder continuity of f α in time.
where γ(r) := exp q (rv), v := exp −1 q (p), for any r ∈ [0, 1], and ∇ 2 is acting on the first variable of p N . Now The first term we bound as where we used (10) together with Lemma 41. 4 The second term we bound as where we used Lemma 41 and the Hölder continuity of f α in time.
4 In coordinates, where Γ are the Christoffel symbols. This gives the quadratic factor in |γ(r)| = d(p, q). The blowup in t − s follows from an application of Lemma 41 (i), (ii) to the components here.
Both are satisfied under our assumptions. Hence Homogeneity α + 2 which is satisfied under our assumptions.
Homogeneity 1 As on homogeneity 0, we write p = p N + R N . We only treat the term involving p N .
It is enough to bound this expression acting on X ∈ T p M . Write For s ∈ [t − d(p, q) 2 , t] we bound (• denotes the dummy variable on which X is acting, · denotes the dummy variable in the distribution-pairing) where we used (10) together with Lemma 41 (ii), as well as the Hölder continuity of f α in time.
where we used (10) together with Lemma 41 (ii) with Y p := d| p exp −1 q (z) X , as well as the Hölder continuity of f α in time.
Both are satisfied under our assumptions.
Consider now s ∈ [0, t−d(p, q) 2 ]. Again it is enough to bound the term acting on some X ∈ T p M . For notational simplicity let v(z) := d| z p N t−s (z, ·) d| p exp −1 z X and ζ s p = R s f (s) − f α (s, p)ξ. We then write the term to bound as where we used (10) together with Lemma 41 (iii). Similarily where we used Lemma 41 (iii) and the Hölder continuity of f α in time. Finally where we used Lemma 41 (ii) and the Hölder continuity of f α in space (9).
Both are satisfied under our assumptions. Then

Time regularity
As on homogeneity 0 we write p = p N + R N . We only treat the term involving p N .
, R r f (r) dr.

Now using (11) and Lemma 41 (i)
Further, again using (11) and Lemma 41 (i) We then needγ Both are satisfied under our assumptions. Then We used the following lemmas.
Lemma 39. Let ρ 1 , ρ 2 ∈ R, g : R 2 → R and assume The following result on heat kernel asymptotics is classical and its proof can be found for example in [D,Theorem 3.10] and [BGV92, Theorem 2.30] See also [Ros97, Section 3.2]. In these references the norm || · || C ℓ (M ×M ) is defined via a partition of unity as in Definition 10. There is a slight difference to our notation. In the cited references, C 1 for example means "continuously differentiable", while in our notation it only means "Lipschitz continuous". But it is enough to know that our norm is dominated by the norm in the references.
Theorem 40. Let M be a d-dimensional, closed Riemannian manifold and p be the heat kernel on M . Then there exist smooth functions (Φ i (p, q)) i≥0 such that if we define for N ≥ 1 Here Φ i (p, q) = 0, for d(p, q) ≥ δ/4.

Lemma 41. Let
Let p ∈ M and define for z in the range of exp −1 p , Y p ∈ T p M a tangent vector and Z ∈ Γ(T M ) a vector field (Note that because of the small support of p N , these are globally well-defined smooth functions by continuation with zero outside of the range of exp −1 p .) Then for any multiindex k, any n ≥ 0 and ℓ = 0, 1.
Proof. The summands of p N are of the same form, apart from the factors t i , i = 0, . . . , N . Since for i ≥ 1 they improve the singularity at t = 0, it is enough to treat N = 0.
Since z → Φ(p, exp p (z)) is smooth, uniformly in p, with support in B 1 (δ/8), and the factor 1/4 in the exponential is irrelevant, we consider where we abuse notation and keep the same name. Now this is the Schwartz function z → exp(−z 2 ) scaled by a factor of √ t, and so part (i) with ℓ = 0 follows from by Remark 9.

Now
The first term is treated as above, now having the additional prefactor t −1 = √ t −2 .
We write the second term as where φ(s) := s 2 exp(−s 2 ) is Schwartz. By Remark 9 part (i) with ℓ = 1 is proven.
For the second statement The first term has worse blowup in t and the factor 1/4 in the exponential is irrelevant, so it is enough to consider f (z)g(z) where Now for a multiindex k .
The third statement follows in a similar fashion from Lemma 42 and Lemma 43.
Lemma 42. Let Y p ∈ T p M acting on the first component of d 2 as follows Then |g(z)| |z||Y | |D β g(z)| |Y |, for any multiindex β.
Lemma 43. Let Then, for any multi-index β, Proof. This follows from the fact that (p, q) → d 2 (p, q) is smooth.
Lemma 44. For any multiindex k Proof. This can be verified using the Faa di Bruno formula.
Indeed, by Theorem 35 and Lemma 45, for a constant c > 0 possibly changing from line to line, since α > −4/3. Hence for T small enough and N large enough, Φ(B(R, N)) ⊂ B(R, N), for any R > 0.
Let us show that Φ is a contraction on B(R, N): for any f, f ′ ∈ B (R, N), Hence for T small enough and N large enough, Φ is a contraction on B(R, N) for any R > 0.
We therefore get unique existence of a solution for small T > 0.

Appendix -The Gaussian model
Let ξ be white noise on M . We recall that ξ is a Gaussian process associated to the Hilbert space L 2 (M, vol M ), on a probability space (Ω, B, P).
Proof. For any coordinate chart ψ defined on an open subset U ⊂ M, and ρ a positive function with support in U , ξ U = ρ • ψ −1 ψ * ξ is a Gaussian process associated to the Hilbert space L 2 (R 2 , ρ 2 • ψ −1 det(g • ψ −1 )). Note that ξ U has the same law as ην, with η := ρ • ψ −1 g • ψ −1 and ν a white-noise on R d . According to [Hai14,Lemma 10.2] ν has a version which is almost surely in C α (R d ) and hence ξ U ∈ C α (R d ).
Let now (ρ i ) 1≤i≤n be a partition a unity subordinated to an atlas (U i , ψ i ) 1≤i≤n . Then, there is a realization of (ξ U i ) 1≤i≤n such that almost surely for all α < −1, i ∈ {1, . . . , n}, ξ U i ∈ C α (R 2 ). Then, n i=1 ψ * i ξ U i is a realization of ξ belonging almost surely to C α (M ).
Thanks to this realization, we can already define the transport map used in the following Lemma (point (i)).
Define the regularity structure and model (in the stronger sense of [Hai14]) where for any s > 0 and ϕ ∈ C ∞ (M ), Note that for any s > 0, ξ ⋄ P s ξ = P s (ξ)ξ − q s .
For any t ≥ 0, let us consider the operator K t = t 0 P s ds and for any p, q ∈ M with p = q, set k t (p, q) = t 0 p s (p, q)ds. Let us note that the operator has a continuous kernel according to Theorem 40, that we shall denote k 2,t .
Proposition 49. For any t ∈ R ≥0 , almost surely for any p ∈ M and ϕ ∈ C ∞ (M ), Z t p , ϕ is well-defined and there exists a modification of the process given by ( Z t p , ϕ ) p∈M,ϕ∈C ∞ (M ),τ ∈G,t≥0 such that almost surely (13) holds true. 6 Proof of Proposition 49. It is enough to prove the assumption of Lemma 48. Let us fix p ∈ M, 0 < r < t. Recall that δ is the radius of injectivity of M and let r := 1.
Let us first check that for any ϕ ∈ C ∞ (M ), Z t x (ϕ), is well defined. Therefor, let us recall -see Theorem 40 -that L := sup The Wick formulas imply for any s > 0, It follows that Var( ξ, ϕP s ξ ) 1/2 ds < ∞.

Higher order covariant derivatives
We want to mirror as best we can the flat space polynomial model described above, in the general context of a d dimensional Riemannian manifold. In order to do to this we need to store higher order derivatives of functions f : M → R in a coordinate independent fashion. There is a canonical way to do this on a Riemannian manifold by making use of the associated Levi-Civita connection.
We recall the notion of higher order covariant derivatives of functions f : M → R on a Riemannian manifold with Levi-Civita 9 connection ∇ (see for example [Lee06,Lemma 4.6]).
and then inductively by; where X 1 , . . . , X ℓ are arbitrary vector fields on M.
A few remarks are in order.
1. As the notation suggests, ∇ ℓ | p f is indeed tensorial, i.e. the right side of the previously displayed equation really only depends on the vector fields, {X i } ℓ i=1 , through their values at p.
2. In the literature ∇f sometimes denotes the gradient of f . We never use the gradient of a function in this work.
3. We shall also sometimes write ∇ ℓ W f = ∇ ℓ | p f, W for any W ∈ (T p M ) ⊗ℓ .
Lemma 53. If f is an ℓ-times continuously differentiable function in a neighborhood of p ∈ M, v ∈ T p M, and γ v (t) := exp p (tv), then More generally, if ℓ, n ∈ N 0 , f is an (ℓ + n + 1)-times continuously differentiable function in a neighborhood of 9 In general, ∇ can be any affine connection.
Proof. Let γ v (t) := exp p (tv) so that γ v (t) solves the geodesic differential equation, ∇γ v (t) /dt = 0 withγ v (0) = v. The proof is completed by showing (by induction) that The case k = 1 amounts to the definition that For the induction step we have by the product rule; wherein the last equality we have again used the product rule to conclude that ∇ dt γ v (t) ⊗k = 0. The result now follows by evaluating (26) at k = ℓ and t = 0. The more general assertion in (25) is proved similarly. One only need observe that ∇W t /dt = 0 and hence the presence of W t in the expressions in no way changes the computations.
Definition 54 (Symmetrizations). If V is a real vector space and ℓ ∈ N, we let Sym ℓ : V ⊗ℓ → V ⊗ℓ denote the symmetrization projection uniquely determined by where S ℓ is the permutation group on {1, 2, . . . , ℓ} . Often we will simplyt write Sym for Sym ℓ as it will typically be clear what ℓ is from the argument put into the symmetrization function.
As usual we let V * denote the dual space to a vector space V and let ·, · denote the pairing between a vector space and its dual. We will often identify (V * ) ⊗ℓ with V ⊗ℓ * where the identification is uniquely determined by We also identify (V * ) ⊗ℓ with the space of multi-linear maps from V ℓ → R using, Under these identification we have and therefore, This formula shows that the symmetric part Sym [T ] of T is completely determined by the knowl- Definition 56. Let Σ ℓ T * p M denote the symmetric tensors in T * p M ⊗ℓ and for T ∈ T * p M ⊗ℓ , let Sym[T ] ∈ Σ ℓ T * p M denote the symmetrization of T as above.
Example 57. If U is an open subset of M and f is ℓ-times continuously differentiable on U , then Sym ∇ ℓ f defines a local section (over U ) of Σ ℓ T * M. Moreover since v ⊗ℓ is symmetric for all v ∈ T p M we may write (24) as Theorem 58 (Taylor's Theorem on M ). Let ℓ, n ∈ N 0 , p ∈ M, v ∈ T p M, γ v (t) := exp p (tv), When ℓ = 0 the previous equation is to be interpreted as (also see [DS15, Theorem 6.1]) Proof. Let g (t) := ∇ ℓ Wt f and recall that the standard Taylor's theorem with remainder states; The results now follow by using Lemma 53 in order to compute the g (k) (t) for 1 ≤ k ≤ n + 1.
Theorem 58 has the following immediate corollaries.
Definition 62 (Taylor approximations). Suppose that U ⊂ M is an open subset of M, p ∈ U, f a n-times continuously differentiable function on M and ε > 0 is sufficiently small so that B p (ε) ⊂ U and ε is smaller than the injectivity radius of M. We then define, Tay n p f ∈ C ∞ (B p (ε)) by With this notation we have the following local version of Corollary 61.
Corollary 63. If f is a n-times continuously differentiable function on M , p, q ∈ M with d (p, q) smaller than the injectivity radius of M, then Remark 64. In the case M = R d and f is a polyonmial of degree at most n, it follows by Taylor's theorem that f = Tay n p f for all p ∈ R d . So in the flat case the error term in (35) is no longer present.
Lemma 65. If f is a n-times continuously differentiable function on M and f (q) = o (d (p, q) n ) , then (V f ) (p) = 0 for any n th -order differential operator V and in particular, ∇ k | p f = 0 for all 0 ≤ k ≤ n.
Proof. Let (Ψ, U ) be a chart on with p ∈ U and Ψ (p) = 0 and define F := f • Ψ −1 ∈ C n Ũ := Ψ (U ) . Then the give assumption implies F (x) = o (|x| n ) and therefore for any x ∈ R d and t ∈ R small we have F (tx) = o (t n ) from which it easily follows that As D k F (0) is symmetric and x ∈ R d was arbitrary we may conclude that D k F (0) = 0 ∈ R d * ⊗k for 0 ≤ k ≤ n. As any n th -order differential operator U on C n (M ) may be written locally as Corollary 66. If f a n-times continuously differentiable function on M and V is an n th -order differential operator, then (V f ) (p) = V Tay n p f (p) and in particular, ∇ n | p f = ∇ n | p Tay n p f (p) from which it follows that ∇ n | p f is a linear combination of Sym ∇ k | p f n k=0 .
Proof. Let D = D x and Γ be the End (T U ) -valued connection one form on T U so that ∇ = D + Γ. It is enough to verify that (36) holds on a basis for T p U ⊗n . To this end, let i j ∈ {1, 2 . . . , d} , for 1 ≤ j ≤ n and let V j = ∂ ∂x i j . Then, which shows that (36) holds for n = 1. For the sake of completing the proof by induction, let us now assume that (36) holds at level n − 1 and below. In particular we assume On one hand, while on the other hand (using the induction hypothesis, the product rule, and DV k = 0 for all k), Comparing the last two displayed equations shows, From this expression it follows that ∇ n Vn⊗···⊗V 1 f may be expressed in the form claimed in (36).
Corollary 71. Let us continue the notation in Lemma 70. Then, there exists Q ℓ,n ∈ Γ Hom T U ⊗n , T U ⊗ℓ , for 1 ≤ ℓ ≤ n, such thatQ n,n = I and for all n-times continuously differentiable functions f, Sym ∇ ℓ f,Q ℓ,n W , ∀ W ∈ T M ⊗n .
Proof. The proof is again by induction on n. For n = 1, we have ψ α Q xα ℓ,n .
We note the following corollary for completeness.

Using
(41) recursively then shows there exists Q ∇ ℓ,n ∈ Γ Hom T M ⊗n , T M ⊗ℓ such that The definitions have been arranged so that Π q τ and Π p Γ p←q τ agree at p to order n, i.e. Π q τ and Π p Γ p←q τ along with all derivatives up to order n agree at p.
Lemma 82. Let τ ∈ T | q and p, z ∈ U where U is a sufficiently small neighborhood of q. If V is a differential operator of order k ≤ n defined on U, then Example 85 (Parallel translation and parallelisms). One natural example of a parallelism when (M, g) is a Riemannian manifold and E is equipped with a covariant derivative, ∇ E , is to define where p, q ∈ M are "close enough" so there is a unique vector v p with minimum length such that q = exp ∇ p (v p ) and // E (·) denotes the parallel translation operator on E relative to ∇ E . For our purposes below E will be a bundle associated to T M and ∇ E will be the induced connection on this bundle associated to the Levi-Civita covariant derivative on (M, g) .
Example 86 (Charts and parallelisms). Each chart (Ψ, U ) induces a local parallelism on (T * M ) ⊗ℓ for any ℓ ∈ N as follows. If A ∈ (T * p M ) ⊗ℓ is expressed as A i 1 ,...,i ℓ dΨ i 1 | p ⊗ · · · ⊗ dΨ i ℓ | p , then we define U Ψ (q, p)A ∈ T * q M ⊗ℓ by In other words, U Ψ (q, p) is uniquely determined by requiring for all q ∈ U and 1 ≤ i 1 , i 2 , . . . i ℓ ≤ d. [This example is basically a special case of Example 85 where one takes ∇ to be the flat connection, D Ψ , defined in Notation 68.] With the aid of a parallelism, we can now define the notion of γ -Hölder section, S, on E. In what follows we assume that E is equipped with a smoothly varying inner product, ·, · E . We do not necessarily assume that ∇ E is compatible with ·, · E or that U (p, q) is unitary for all (p, q) ∈ D.
Proof. We work in a local trivialization. Let U, U ′ : R d ×R d → GL R N be smooth functions such that U (x, x) , U ′ (x, x) = I, which we view to be a parallelism on the trivial bundle, R d × R N over R d . A continuous section of this bundle may be identified with a continuous function, S : R d → R N Then ||U (x, y)S(y) − S(x)|| ≤ || U (x, y) − U ′ (x, y) S(y)|| + ||U ′ (x, y)S(y) − S(x)||.
The statement then follows from smoothness of U, U ′ , the fact that they coincide at x, x and local boundedness of S.
Lemma 88. Let f ∈ C (M ) , γ > 0 and n = ⌊γ⌋ ∈ N 0 . Then f ∈ C γ (M ) (as in Definition 10) iff f a n-times continuously differentiable function on M and for any (local) parallelisms (U ) on the vector bundle Σ n T * M, Sym[∇ n |f ] satisfies where ∇ is the Levi-Civita covariant derivative.
Proof. Recall from Definition 10, that f ∈ C (M ) is in C γ (M ) iff f • Ψ −1 ∈ C γ (Ψ(U )) for every coordinate chart (Ψ, U ). These conditions are equivalent to f being n-times continuously differentiable and the n th -derivatives of f • Ψ −1 are locally (γ − n)-Hölder on Ψ (U ) . The latter condition may be expressed as saying where L is a linear differential operator of order at most n−1. As Lf is continuously differentiable it follows that (q, p) → U Ψ (q, p) (Lf ) p − (Lf ) q is continuously differentiable and vanishes at q = p and therefore (by the fundamental theorem of calculus) U Ψ (q, p) (Lf ) p − (Lf ) q d(q, p).
From (46) and (47) it follows that (45) is equivalent to Lastly using Lemma 87 we conclude that the estimates in (48) and (44) are also equivalent.
Theorem 89. Fix n ∈ N and construct T and (Π, Γ) as above. Then T is a regularity structure and (Π, Γ) is a model of transport precision n + 1.
Let D be the covariant derivative induced by the chart exp −1 q . Using Lemma 70 we get and hence |Γ p←q τ ℓ | m d(p, q) ℓ−m , which finishes the proof.
We are finally able to characterize C γ (M ) in terms of the "polynomial" regularity structure. and since the expression is smooth in q we can focus on | Sym ∇ n | q f − ∇ n | q ∇ n | p f, exp −1 p (·) ⊗i | Define on the vector bundle Σ n T * M the parallelism U (q, p)S := ∇ n | q S, exp −1 p (·) ⊗n .
Step 1: We will show that f is n-times differentiable and Sym[∇ ℓ f ] =f ℓ for ℓ = 0, . . . , n. This will be done by induction.