Abstract
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.
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1 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
Here \(u: [0,T] \times D \rightarrow \mathbb {R}\) is looked for, where D is some 2 dimensional domain, and \(\xi \) is (time-independent) 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 \(u \xi \) to be well-defined analytically. The standard tool of stochastic calculus, the Itō integral, is also of no use here, since the white-noise is constant in time.
With the breakthrough results of Hairer [9] and Gubinelli, Imkeller and Perkowski [7] a large class of such equations has become amenable to analysis. Let us sketch the approach of [9], since this is the one we shall use in this work.
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assume that u “looks like” the solution \(\nu \) to the additive-noise equation
$$\begin{aligned} \partial _t \nu = \mathop {}\mathbin \bigtriangleup \nu + \xi , \end{aligned}$$(1)which is classically well-defined via convolution with the heat semigroup \(P_t\)
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under this assumption, if we somehow define\(\nu \cdot \xi \), then the framework defines \(u \cdot \xi \) automatically
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close the fixpoint argument, i.e.
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1.
u “looks like” \(\nu \)
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2.
\(w := P_t u_0 + \int _0^t P_{t-s} [ u_s \xi ] ds\)
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3.
then w “looks like” \(\nu \)
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1.
It then only remains to define the missing ingredient “\(\nu \cdot \xi \)”. 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 [9] that (PAM) possesses a unique solution for \(D = \mathbb {T}^2\), 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 \((\mathbb {R}^d,+)\) 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 21 and in the construction of the Gaussian model in Sect. 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 everything is laid out explicitly and covers the flat case \(M = \mathbb {T}^2\), 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 [2], 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 [3] though, which pushes the framework to more general equations.
The outline of this paper is as follows. After presenting notational conventions, we give in Sect. 2 the notion of distributions on manifold we shall use in this work. Moreover we introduce Hölder spaces on manifolds. In Sect. 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 Sect. 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 Sect. 5. As input it takes the product \(\nu \xi \) alluded to before. This is constructed in Sect. 8 via renormalization. Section 6 gives the Schauder estimate for modelled distributions in the setting of PAM and finally Sect. 7 solves the corresponding fixpoint equation. In Sect. 9 we show how the construction of Sect. 4 can be extended to “polynomials” of arbitrary order.
1.1 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 \(\delta > 0\) the radius of injectivity of M. For \(p \in M\) we denote with \(\exp _p: T_p M \rightarrow M\) the exponential function. It is a diffeomorphism on \(B_{T_p M}(0_p, \delta ) := \{ x \in T_p M : |x|_{T_p M} < \delta \}\), with inverse \(\exp _p^{-1}\).
For a function \(\varphi \) supported in \(B_{T_p M}(0_p, \delta )\) define for \(\lambda \in (0,1], p \in M,\)
extended to all of M by setting it to zero outside of \(B_{T_p M}(0_p, \delta )\).
For \(\tau \in \mathcal {G}\), \(\mathcal {G}\) a graded normed vector bundle with grading A we denote by \(||\tau ||_a\) the size of component in the a-th level, \(a \in A\).
The differential of a smooth enough function \(f: M \rightarrow \mathbb {R}\) at a point p will be denoted \(d|_p f \in T^*_p M\). Similarly for higher order derivatives (see Sect. 9) \(\nabla ^\ell |_p f \in \left( T^*_p M \right) ^{\otimes \ell }\). For the action on vectors \(W \in \left( T_p M \right) ^{\otimes \ell },\) we shall write either \(\langle \nabla ^\ell |_p f, W \rangle \) or \(\nabla _{W} f\).
For \(p\in M\), \(r, \delta > 0\) denote
Here r will depend on the situation, and will always be large enough so that the distributions under consideration can act on \(\varphi \).
We shall use p, q for points in M and x, y, z to denote points in \(\mathbb {R}^d\). For \(x \in \mathbb {R}^d\), \(\varphi : \mathbb {R}^d \rightarrow \mathbb {R}\) we write
which is consistent with the notation introduced above when considering \(\mathbb {R}^d\) as Riemannian manifold with the standard metric. We also define, analogously to above, \(B_{\mathbb {R}^d}(0, \delta )\) and \(\mathcal {B}_{\mathbb {R}^d}^{r,\delta }\). Balls in M are denote by \(B_{M}(p, \delta ) := \{ q \in M : d(q,p) < \delta \}\).
For \(\gamma \in \mathbb {R}\) we denote by \([\gamma ]\) the largest integer strictly smaller than \(\gamma \).
For a pairing of a distribution T with a test function we write \(\langle T, \varphi \rangle \).
For two quantities f, g we write \(f \lesssim g\) if there exists a constant \(C> 0\) such that \(f \le C g\). To make explicit the dependence of C on a quantity h, we sometimes write \(f \lesssim _h g\).
We denote the positive natural numbers by \(\mathbb {N}\) and the non-negative ones by \(\mathbb {N}_{0}\).
2 Hölder spaces
Definition 1
A distribution on M is a bounded, linear functional on \(C^\infty _c( M )\) (\(=C^\infty (M)\), if M is compact).
Given a density \(\lambda \) on M, \(\langle T_\lambda , \varphi \rangle := \int _M \varphi d\lambda \) defines a distribution. Distributions are hence “generalized densities”. Compare [6, Section 2.8] and [17, Section 1.3].
There is another definition of distributions as “generalized functions”, see [12, Section 1.8]. They are equivalent when there is a canonical way to turn a function into a density and vice versa. This is the case when there is a reference density, like on a Riemannian manifold.
Remark 2
On a Riemannian manifold M, denote the standard density by \(d\,{\text {|Vol|}}\). We can lift a function \(f \in C(M)\) to a density \(f d\,{\text {|Vol|}}\). Then, for \(f \in C^\infty _c(M)\), \(T_f\) defined as
is a distribution.
Definition 3
(Push-forward) Let \((\Psi , \mathcal {U})\) be a coordinate chart on M. If \(\varphi \in C^\infty _c( \Psi (\mathcal {U}) )\) and T is a distribution on M we can define the push-forward \(\Psi _* T \in \mathcal {D}'(\Psi (\mathcal {U}))\) via
Remark 4
This push-forward is compatible with the pull-back of densities. Indeed, for \(f \in C(M)\) we get the distribution \(T_f := f d\,{\text {|Vol|}}\), by Remark 2. This density pulls back under \(\Psi ^{-1}\) as (compare [13, Proposition 16.38])
where \(y^i\) are standard coordinates on \(\mathbb {R}^d\), and g is the Riemannian metric in the coordinates \(\Psi \). Hence
where the last line is the pairing of a distribution with a test function on \(\mathbb {R}^d\) and \(\bar{T}_h\) is the canonical identification of a locally integrable density h on \(\mathbb {R}^d\) with a distribution.
Recall the following definition of Hölder spaces in Euclidean space.
Definition 5
For \(\gamma > 0\) we keep the classical definition, that is
where \(\gamma = n + s\), \(n \in \mathbb {N}_{0}\), \(s \in (0,1]\).
For \(\gamma \le 0\) denote by \(C^\gamma (\mathbb {R}^d)\) the space of distributions \(T \in \mathcal {D}'(\mathbb {R}^d)\) with
Here \(r := - [\gamma ]\), and \(\varphi ^\lambda _x\) as well as the set of test functions \(\mathcal {B}_{\mathbb {R}^d}^{r,1}\) are defined in Sect. 1.1.
Remark 6
For \(\gamma < 0\) the norm is independent of the arbitrary upper bound 1 for the supremum over \(\lambda \) as well as the support of \(\varphi \): for every \(\lambda _0, \varepsilon _0 > 0\)
where \(r := - [\gamma ]\).
Remark 7
Every time that a condition like
appears, for some \(\varepsilon > 0\), uniformly over \({\text {supp}}\,\varphi \subset B_{\mathbb {R}^d}(0, \varepsilon ),\) with \(||\varphi ||_{C^r(B_{\mathbb {R}^d}(0, \varepsilon ))} \le 1,\) one can equivalently demand for some \(\varepsilon > 0\)
uniformly over \(\lambda \in (0,1]\) and \({\text {supp}}\,\varphi \subset B_{\mathbb {R}^d}(x, \lambda \varepsilon )\), with \(||D^k \varphi ||_\infty \lesssim \lambda ^{-d - k}\), for \(k=0,\dots ,r\).
We need a reformulation similar to this remark, but for Schwartz test functions.
Lemma 8
Let \(\gamma \le 0\) and \(T \in C^\gamma (\mathbb {R}^d)\). Then \(T \in S'(\mathbb {R}^d)\) (and not just \(T \in \mathcal {D}'(\mathbb {R}^d)\)). Moreover, define for \(\varphi \in S(\mathbb {R}^d), \lambda \in (0,1], x_0 \in \mathbb {R}^d, N \in \mathbb {N}\) and \(r := -[\gamma ]\)
Then, for \(N > d\),
Remark 9
Note that if \(\varphi \in S(\mathbb {R}^d)\), then \(\varphi ^\lambda _{x_0} := \lambda ^{-d} \varphi (\lambda ^{-1}( \cdot - x_0) )\) satisfies for \(\lambda \in (0,1]\), \(N \in \mathbb {N}\), \(x_0 \in \mathbb {R}^d\) and \(r > 0\),
Proof
Let \(\phi _z\), \(z \in \mathbb {Z}^d\), be a partition of unity of \(\mathbb {R}^d\) such that \({\text {supp}}\,\phi _z \subset B_{\mathbb {R}^d}(z, 1)\) and \(\sup _{z \in \mathbb {Z}^d} ||\phi _z||_{C^r} < \infty \). Define
Then \(\sum _{z \in \mathbb {Z}^d} \varphi _{z,\lambda } = \varphi \). Write for short \(C_\varphi := C(\varphi ,\lambda ,x_0,N,r)\). We have \({\text {supp}}\,\varphi _{z,\lambda } \subset B_{\mathbb {R}^d}(\lambda z, \lambda )\) and, since \(\varphi \in S(\mathbb {R}^d)\),
Then
as desired. We used the fact that \(\sum _{z \in \mathbb {Z}^d} \frac{1}{1 + |z - \lambda ^{-1} x_0|^{N}}\) is upper bounded by \(\int _{\mathbb {R}^d} \frac{1}{1 + |z - \lambda ^{-1} x_0|^{N}} dz = \int _{\mathbb {R}^d} \frac{1}{1 + |z|^{N}} dz\), which is finite, since \(N>d,\) and independent of \(\lambda \). \(\square \)
Definition 10
Let M be a closed Riemannian manifold. Let a finite partition of unity \((\phi _i)_{i\in I}\) be given on M, subordinate to a finite atlas \((\Psi _i, U_i)_{i\in I}\). For \(\gamma \in \mathbb {R},\) define
and
For \(\gamma > 0,\) an equivalent characterization of \(C^\gamma (M)\) will be shown in Theorem 92. We now give one in the case \(\gamma \le 0\).
Lemma 11
For \(\gamma \le 0\), M a closed Riemannian manifold, an equivalent norm on \(C^\gamma (M)\) is given by
where we recall that \(\varphi ^\lambda _p\) is defined in (2).
Proof
Fix a finite atlas \((\Psi _i, U_i)_{i \in I}\) with subordinate partition of unity \(\phi _i\). Denote
(\(C_2 \lesssim C_1\)): Let \(\varphi \in \mathcal {B}_{T_p M}^{r,\delta }\), \(p \in M\). Then
with \(\eta _i := (\tilde{\phi }_i \varphi ^\lambda _p) \circ \Psi ^{-1}_i\). Here the functions \(\tilde{\phi }_i\) are chosen such that \({\text {supp}}\,\tilde{\phi }_i \subset U_i\) and \(\phi _i \tilde{\phi _i} = \phi _i\). Now, \(\varphi ^\lambda _p\) is supported in a ball of radius \(\delta \lambda \) around p, hence for some \(c > 0\) independent of i
for some \(z_i \in \mathbb {R}^d\). Here, the last inclusion follows from the fact that the atlas is finite and \(\tilde{\phi }_i\) is strictly contained in \(U_i\), so that the Lipschitz norm of \(\Psi _i\) and \(\Psi _i^{-1}\) are uniformly bounded in the regions of interest. By the same reasoning, the derivatives up to order r of \(\exp ^{-1}_p \circ \Psi ^{-1}_i\) are uniformly bounded on the relevant regions. Hence
and the result follows from Remark 7.
(\(C_1 \lesssim C_2\)): By Remark 6, we have to show for some \(\lambda _0 >0\) and for all \(i \in I\), \({\text {supp}}\,\varphi \subset B_{\mathbb {R}^d}(0, 1)\), \(||\varphi ||_{C^r} \le 1\) and \(x \in \mathbb {R}^d\)
Since \(\phi _i\) is supported away from the boundary of \(U_i\), there exists \(\lambda _0 > 0\) such that for all \(x \in \mathbb {R}^d\) there is some \(p \in M\) such that \({\text {supp}}\,\varphi _{i;\lambda _0;x} \subset B_{M}(p, \delta )\).Footnote 1 Since the atlas is finite, \(\lambda _0\) can be chosen uniformly for all \(i \in I\).
Then, for \(\lambda \le \lambda _0\)
Now, one checks that \(\varphi _{i;\lambda ;x} \circ \exp _{\Psi ^{-1}_i(x)}\) falls under Remark 7 and hence this expression is indeed bounded by a constant times \(C_2 \lambda ^\gamma \). \(\square \)
As immediate consequence we get the following statement.
Corollary 12
Let \((\bar{\Psi }_j, \bar{U}_j)_{j\in J}\) be another finite atlas with subordinate partition of unity \((\bar{\phi }_j)_{j\in J}\). Then for \(\gamma \le 0\)
with equivalent norms.
3 Regularity structures on manifolds
Let M be a d-dimensional Riemannian manifold without boundary. The two cases we are most interested in are
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M is compact without boundary (i.e. closed)
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M is an open bounded subset of \(\mathbb {R}^d\) with induced Euclidean metric
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 Sect. 4 for the implementation of a first order “polynomial” structure; to Sect. 9 for a structure implementing “polynomials” of any order and right before Lemma 36 for the structure used for the parabolic Anderson model.
Definition 13
(Regularity structure) A regularity structure is a graded vector bundle \(\mathcal {G}\) on M, with a finite grading \(A = A(\mathcal {G}) \subset \mathbb {R}\). For \(a \in A\), \(\mathcal {G}_a\) denotes the vector bundle of homogeneity a; it is assumed to be finite dimensional. We denote the fiber at \(p \in M\) by \(\mathcal {G}|_p\) and the fiber of homogeneity a at p by \(\mathcal {G}_a|_p\). For \(p \in M, \tau \in \mathcal {G}|_p\), \(a \in A\) we write \({\text {proj}}_{\mathcal {G}_a|_p} \tau \) for the projection of \(\tau \) onto \(\mathcal {G}_a|_p\). We assume that \(\mathcal {G}\) comes equipped with a norm ||.||. We denote the fiber norm of the restriction to homogeneity a by \(||\tau ||_a := ||{\text {proj}}_{\mathcal {G}_a|_p} \tau ||\).
Definition 14
(Model) Let a collection of open sets \(\mathcal {U}_q \subset M\), \(q \in M\), with \(q \in \mathcal {U}_q\), and linear maps
be given. We assume there is for every compactum \(\mathcal {K}\subset M\) a constant \(\delta _\mathcal {K}= \delta _\mathcal {K}(\Pi ,\Gamma ,\{\mathcal {U}_q\}_q) > 0,\) such that \(\Gamma _{p \leftarrow q}\) is defined for \(p,q \in \mathcal {K}, d(p,q) < \delta _\mathcal {K}\) and for \(q \in \mathcal {K}\), \(\exp _q|_{B_{T_p M}(0_p, \delta _\mathcal {K})}\) is a diffeomorphism and \(\exp _q(B_{T_p M}(0_p, \delta _\mathcal {K})) \subset \mathcal {U}_q\). Assume moreover \(\Gamma _{p \leftarrow p} = {\text {id}}\) for every \(p \in M\). Given \(\beta \in \mathbb {R}\), we say that \((\Pi , \Gamma )\) is a model with transport precision\(\beta \) if the following entity is finite for every compactum \(\mathcal {K}\subset M\)
where we recall that the set of test functions \(\mathcal {B}_{T_p M}^{r,\delta }\) was defined in (3).
Remark 15
The additional restriction on distance and support in the second supremum are necessary, since otherwise the action of \(\Pi _q \tau \) on \(\varphi _p^\lambda \) might not be well-defined.
Remark 16
Note that the conditions on a model do not pin down the global regularity of \(\Pi _q \tau \). Without loss of generality we will assume that \(\Pi _q \tau \in C^\alpha (U_q)\) for all \(q \in M, \tau \in \mathcal {G}|_q\) and \(\alpha := \min A(\mathcal {G})\).
Our definition of a regularity structure and a corresponding model are slightly more general than the original formulation by Hairer [9]. This extension is necessary to accommodate the “polynomial regularity structure”, which will be constructed up to first order in Sect. 4 and up to any order in Sect. 9. Let us point out the key differences.
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Derivatives of functions on a general manifold M can only be coordinate invariantly stored in a fibered space. Hence the regularity structure has to be a vector bundle and not a fixed vector space.
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For this reason there cannot be a fixed structure group G in which the transport maps \(\Gamma _{p \leftarrow q}\) take value.
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The transport maps \(\Gamma _{p \leftarrow q}\) can also act “upwards”, see Remark 83.
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The distributions \(\Pi _p \tau \) as well as the transports \(\Gamma _{p \leftarrow q}\) only make sense locally.
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The identities \(\Gamma _{p \leftarrow q} \Gamma _{q \leftarrow r} = \Gamma _{p \leftarrow r}\) and \(\Pi _p \Gamma _{p \leftarrow q} = \Pi _q\) do not hold. An approximate version of the latter is incorporated into the norm (transport precision \(\beta \)). The former is, in the flat case, used in an extension argument ([9, Proposition 3.31]), which we do not need here.
It turns out that the theory can handle these slight extensions. In particular the reconstruction theorem still holds, Theorem 23. 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 18) on a manifold.
As in Lemma 8 we know how \(\Pi _p \tau \) acts on a more general class of functions:
Lemma 17
For a regularity structure \(\mathcal {G}\) let be given a model \((\Pi ,\Gamma )\) of transport precision \(\beta \) with \(\beta \ge \sup _{a \in A(\mathcal {G})} |a|\). Let \(p \in \mathcal {K}\), a compactum in M. Let \(\lambda \in (0,1]\) and let \(\varphi \) satisfy the assumptions of Lemma 8, with \(\mathbb {R}^d\) replaced by \(T_p M\) and the additional condition \({\text {supp}}\,\varphi \subset B_{T_p M}(0_p, \delta _\mathcal {K}/ 4)\). Assume moreover that \(B_{M}(p, \delta _\mathcal {K}/2) \subset \mathcal {K}\) (which can always be achieved by making \(\delta _\mathcal {K}\) smaller). Then for \(\tau \in \mathcal {G}_\ell |_p\)
where \(C_\varphi := C(\varphi ,\lambda ,0,N,r)\) is defined in Lemma 8.
Proof
Let \(O: \mathbb {R}^d \rightarrow T_p M\) be a linear isometry. Uniform estimates on \(D^k \varphi \) on balls in \(T_p M\) are equivalent to uniform estimates on \(D^k (\varphi \circ O)\) on balls in \(\mathbb {R}^d\). We can hence consider \(\varphi \) as a function on \(\mathbb {R}^d\).
Let \(\phi _z\), \(z \in \mathbb {Z}^d\), be a partition of unity of \(\mathbb {R}^d\) such that \({\text {supp}}\,\phi _z \subset B_{\mathbb {R}^d}(z, 1)\) and \(\sup _{z \in \mathbb {Z}^d} ||\phi _z||_{C^r} < \infty \). Let \(\lambda _\mathcal {K}:= \lambda \delta _\mathcal {K}/ 4\). Define
so that
Then \({\text {supp}}\,\varphi _{z,\lambda _\mathcal {K}} \subset B_{\mathbb {R}^d}(\lambda _\mathcal {K}z, \lambda _\mathcal {K}) \cap B_{\mathbb {R}^d}(0, \delta _\mathcal {K}/4)\). Hence \(\varphi _{z,\lambda _\mathcal {K}} \equiv 0\) for \(|\lambda _\mathcal {K}z| \ge \delta _\mathcal {K}/ 2\). Moreover
Then
Note that in the sum \(|\lambda _\mathcal {K}z| < \delta _\mathcal {K}/ 2\). Hence, by assumption \(q := \exp _p(\lambda _\mathcal {K}z ) \in \mathcal {K}\). Hence by definition of a model, \(||\Gamma _{p \leftarrow q} \tau ||_m \le ||\Pi ,\Gamma ||_{\beta ,\mathcal {K}} ||\tau || d(p,q)^{(\ell -m)\vee 0} = ||\Pi ,\Gamma ||_{\beta ,\mathcal {K}} ||\tau || |\lambda _\mathcal {K}z|^{(\ell -m)\vee 0}\), for \(\tau \in \mathcal {G}_\ell |_q\).Footnote 2 Then for those z
Moreover, the model being of transport precision \(\beta \), we get
Combining,
\(\square \)
Definition 18
Let \(\mathcal {G}\) be a regularity structure and \((\Pi ,\Gamma )\) a model of precision \(\beta \in \mathbb {R}\). Define for \(\gamma > \sup _{\alpha \in A(\mathcal {G})} |\alpha |\) the space of modelled distributions
with
Here \(\delta _\mathcal {K}\) is the distance of points in \(\mathcal {K}\) for which \(\Gamma \) makes sense, see Definition 14. Note that the precision of transport \(\beta \) plays no role here.
Remark 19
As usual for Hölder norms, for every compactum \(\mathcal {K}\) an equivalent norm is obtained by replacing in the supremum, for any \(\delta ' \in (0,\delta _\mathcal {K}]\), the condition \(d(p,q) < \delta _\mathcal {K}\) with the condition \(d(p,q) < \delta '\).
Lemma 20
(Push-forward) Let M, N be Riemannian manifolds and let \(\Psi : M \rightarrow N\) be a diffeomorphism. Let \(\mathcal {G}\) be a regularity structure on M with model \((\Pi ,\Gamma )\) with transport precision \(\beta \in \mathbb {R}\). Define
Then, \(\bar{\mathcal {G}}\) is a regularity structure on N with grading \(\bar{}A = A\) and \((\bar{\Pi }, \bar{\Gamma })\) is a model with transport precision \(\beta \). Moreover for every compactum \(\mathcal {C} \subset N\) and all compacta \(\mathcal {K}\subset \mathcal {C}\)
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1.
$$\begin{aligned} ||\bar{\Pi }, \bar{\Gamma }||_{\beta ,\mathcal {K}}&\lesssim _{\mathcal {C}} ||\Pi ,\Gamma ||_{\beta ,\Psi ^{-1}(\mathcal {K})} \end{aligned}$$
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2.
Let \(f, f' \in \mathscr {D}^\gamma ( M, \mathcal {G})\) and define \(\tilde{f}(x) := f( \Psi ^{-1}(x) ), \tilde{f}'(x) := f'(\Psi ^{-1}(x))\). Then \(\tilde{f}, \tilde{f}' \in \mathscr {D}^\gamma (\Psi (\mathcal {U}), \mathcal {G})\) and
$$\begin{aligned} ||\bar{f}||_{\mathscr {D}^\gamma (\mathcal {K}, \mathcal {G})}&\lesssim _{\mathcal {C}} ||f||_{\mathscr {D}^\gamma ( \Psi ^{-1}(\mathcal {K}), \mathcal {G})} \\ ||\tilde{f} - \tilde{f}'||_{\mathscr {D}^\gamma (\mathcal {K}, \mathcal {G})}&\lesssim _{\mathcal {C}} ||f - f'||_{\mathscr {D}^\gamma ( \Psi ^{-1}(\mathcal {K}), \mathcal {G})}. \end{aligned}$$
Proof
Since \(\Psi \) has derivative uniformly bounded below and above for every compactum \(\mathcal {K}\subset \mathcal {C}\), one can choose for every compactum \(\mathcal {K}\) a constant \(\bar{\delta }_\mathcal {K}\) as in the definition of a model, such that \(\bar{\Gamma }_{p \leftarrow q}\) is well-defined for \(p,q \in \mathcal {K}\) and \(d(p,q) < \delta _\mathcal {K}\) as well as \(\exp ^N_q( B_{T_q M}(0_q, \delta _\mathcal {K}) \subset \bar{\mathcal {U}}_q\). Here \(\exp ^N\) denotes the exponential map on N.
1. Let \(q \in \mathcal {K}\subset N\) and \(\tau \in \tilde{\mathcal {G}}_a|_q\) and \(\varphi \in \mathcal {B}_{T_q M}^{r,\delta _\mathcal {K}}\)
since \(\varphi ^\lambda _q \circ \Psi \circ \exp _q\) falls under Remark 7. For \(p,q \in \mathcal {K}\subset N\) with \(d(p,q) < \delta _\mathcal {K}\) and \(\tau \in \tilde{\mathcal {G}}|_q\), we have
again by Remark 7. Finally for \(p,q \in \mathcal {K}\subset N\) with \(d(p,q) < \delta _\mathcal {K}\) and \(\tau \in \tilde{\mathcal {G}}_a|_q\), we have
2. Let \(p,q \in \mathcal {K}\subset N\) then
and similarly for the distance of two modelled distributions. \(\square \)
Lemma 21
(Reconstruction for \(M \subset \mathbb {R}^d\)) Let \(\mathcal {G}\) be a regularity structure on M, an open connected subset of \(\mathbb {R}^d\). Let \((\Pi , \Gamma )\) be a model with precision \(\beta \in \mathbb {R}\). Let \(\gamma > 0\) and assume \(\beta \ge \gamma \). Denote \(\alpha := \inf A\). Assume either that \(\alpha < 0,\) or that \(\alpha = 0\) and that the lowest homogeneity in \(\mathcal {G}\) is given by the constant distribution (of the polynomial regularity structure of Sect. 4).
For every \(f \in \mathscr {D}^\gamma (M, \mathcal {G})\) there exists a unique \(\mathcal {R}f \in C^\alpha (M)\) such that for every compactum \(\mathcal {K}\subset M\)
Here \(\varphi \in \mathcal {B}_{T_p M}^{r,\delta _\mathcal {K}}\), \(r = -[\alpha ]\), (so that the action of \(\Pi _x f(x)\) is well-defined) and \(\overline{\mathcal {K}}\) is the closure of the \(\delta _\mathcal {K}\) thickening of \(\mathcal {K}\).
Remark 22
Uniqueness actually holds in the class of operators \(\mathcal {R}\) that satisfy (5) with \(\gamma \) replaced by any \(\theta > 0\).
Proof
Existence
We will apply [9, Proposition 3.25].Footnote 3 This Proposition is formulated for \(\mathbb {R}^d\), but the statement is local and also holds for \(M \subset \mathbb {R}^d\). So we have to verify for \(\zeta _x := \Pi _x f(x)\)
uniformly over \(x,y \in \overline{\mathcal {K}}\), \(n \ge n_0\), \(n_0 = \log _2( \delta _{\overline{\mathcal {K}}} ) \vee 0\) and \(2^{-n} \le |x-y| \le \delta _{\overline{\mathcal {K}}}\). In [9, Proposition 3.25] the upper bound 1 is chosen on \(|x-y|\), but any upper bound works, so we chose \(\delta _{\overline{\mathcal {K}}}\), since we need \(\Gamma _{x \leftarrow y}\) to be well-defined.
Here
and \(\varphi \) is a scaling function for a wavelet basis of regularity \(r > |\alpha |\). We have chosen \(n_0\) also such that for \(n \ge n_0\) and \(x \in \overline{\mathcal {K}}\), \(\tau \in \mathcal {G}|_x\) the expression \(\langle \Pi _x \tau , \varphi ^n_x \rangle \) is well-defined. First, (7) follows from the fact that \(\alpha \) is the lowest homogeneity in \(A(\mathcal {G})\) (note that \(\varphi ^n_x\) is scaled to preserve the \(L^2\)-norm, whereas the scaling in the definition of a model preserves the \(L^1\)-norm).
Now
We bound the first term as
since \(2^{-n} \le |x-y|\). The second term is bounded as
where we again used the assumption \(2^{-n} \le |x-y|\). This proves (6) and an application of [9, Proposition 3.25] gives the existence of \(\mathcal {R}f\) satisfying the bound (5).
The preceding argument is valid for \(\alpha < 0\). For \(\alpha = 0\), one can run the argument for some \(\alpha ' < 0\) and get unique existence of \(\mathcal {R}f \in C^{\alpha '}\) with the claimed properties. In Corollary 24 below it is shown that actually \(\mathcal {R}f \in C^0\).
Uniqueness
Uniqueness follows exactly as in [9, Section 3]. \(\square \)
Lemma 23
(Reconstruction for M a closed Riemannian manifold) Let M be a closed Riemannian manifold with regularity structure \(\mathcal {G}\) and \((\Pi , \Gamma )\) a model with transport precision \(\beta \in \mathbb {R}\). Let \(\gamma > 0\), and \(f \in \mathscr {D}^\gamma (M,\mathcal {G})\) and assume \(\beta \ge \gamma \).
Denote \(\alpha := \inf A\). Assume either that \(\alpha < 0\) or that \(\alpha = 0\) and that the lowest homogeneity in \(\mathcal {G}\) is given by the constant distribution (of the polynomial regularity structure).
Then, there exists a unique distribution \(\mathcal {R}f \in C^{\alpha }(M)\) such that
for \(p \in M\), \(\varphi \in \mathcal {B}_{T_p M}^{r,\delta _M}\), \(r = - [\alpha ]\).
Proof
By a cutting up procedure, it is enough to show (8) for \(\varphi \in \mathcal {B}_{T_p M}^{r,\delta '}\) with \(\delta ' \in (0,\delta _M]\) to be chosen.
Let \((\Psi _i, \mathcal {U}_i)_{i\in I}\) a finite atlas with subordinate partition of unity \((\phi _i)_{i\in I}\). On each chart, we push-forward the regularity structure, model and f to \(\Psi _i(\mathcal {U}_i),\) with corresponding reconstruction operation \(\tilde{\mathcal {R}}_i\), model \(\tilde{\Pi }_i\) and modelled distribution \(\tilde{f}_i\). For each \(i\in I,\) fix a compactum \(\mathcal {K}_i \subset \mathcal {U}_i\) such that \({\text {supp}}\,\phi _i\) is strictly contained in \(\mathcal {K}_i\). By Lemma 20,
Now reconstruct in each coordinate chart as \(\tilde{T}_i := \tilde{\mathcal {R}}_i \tilde{f}_i\) using Theorem 21. Define \(\mathcal {R}f := \sum _{i\in I} \phi _i (\Psi ^{-1}_i)_* \tilde{T}_i\). Then
If \(p \not \in \mathcal {U}_i,\) we want the summand to vanish. So let \(\delta ' := \min _i d({\text {supp}}\,\phi _i, \partial \mathcal {U}_i)\). Then for \(\varphi \in \mathcal {B}_{T_p M}^{r,\delta '}\), \(\lambda \in (0,1],\) we have \(\phi _i \varphi ^\lambda _p \not = 0\) implies \(p \in \mathcal {U}_i\). Hence, if \(p \not \in \mathcal {U}_i,\) we have \(\phi _i \varphi ^\lambda _p = 0\), so the summand vanishes.
Otherwise, with \(z := \Psi _i(p)\)
since \(\left( \phi _i \lambda ^{-d} \varphi ( \lambda ^{-1} \exp _p^{-1}( \cdot ) ) \right) \circ \Psi _i^{-1}\) falls under Remark 7 around z. Summing over i gives (8). \(\square \)
Corollary 24
In the setting of the previous theorem, assume that the lowest homogeneity in \(\mathcal {G}\) is 0 and that it is given by the constant (as in the polynomial regularity structure of Sect. 4). Then \(\mathcal {R}f\) is given by projection onto that homogeneity, i.e.
Proof
Define \(\tilde{\mathcal {R}} f(p) := f_0(p)\), then
Recall that the projection \({\text {proj}}\) is defined in Definition 13. The last term is of bounded by a constant times \(\lambda ^\eta \), where \(\eta \) is the smallest homogeneity strictly larger than 0.
For the second to last term we first write
Now, since \(f \in \mathcal {D}^\gamma \),
By the properties of a model
Hence \(|f_0(\cdot ) - f_0(p)| \lesssim d(\cdot ,p)^{\eta \wedge \gamma }\) and then
Hence, by Remark 22, \(\tilde{\mathcal {R}} = \mathcal {R}\). \(\square \)
We want to apply the Lemma 23 to the terms in the heat kernel asymptotics (Theorem 43). The problem is that their support will be of order 1 (and not of order \(\lambda \) as for \(\varphi ^\lambda _x\)). Hence we need the following refinement which is similar to Lemma 8.
Lemma 25
In the setting of Lemma 23, let \(\varphi \) satisfy the assumptions of Lemma 8 with the additional condition \({\text {supp}}\,\varphi \subset B_{T_p M}(0_p, \delta _\mathcal {K}/ 4)\). Then
where \(C_\varphi := C(\varphi ,\lambda ,0,N,r)\) is defined in Lemma 8.
Proof
Let \(\varphi _{z,\tilde{\lambda }}\) be given as in the Proof of Lemma 17 with \(\mathcal {K}:= M\).
Recall \(\lambda _M := \lambda \delta _M / 4\), and that \({\text {supp}}\,\varphi _{z,\lambda _M} \subset B_{\mathbb {R}^d}(\lambda _M z, \lambda _M) \cap B_{\mathbb {R}^d}(0, \delta _M/4)\). Hence \(\varphi _{z,\lambda _M} \equiv 0\) for \(|\lambda _M z| \ge \delta _M / 2\). Then with \(\zeta _r := \mathcal {R}f - \Pi _r f(r)\)
Note that in the sum \(|\lambda _M z| < \delta _M / 2\). Hence \(\exp _p(\lambda _M z ) \in M\) is well-defined. Now the first summand can be written as
Applying Remark 7 to \(\varphi _{z,\lambda _M} \circ \exp ^{-1}_p \circ \exp _{\exp _p(\lambda _M z)}\) and (8), this is bounded by a constant times \(C_\varphi ||\Pi ,\Gamma ||_{\beta ,M}\ ||f||_{\mathcal {D}^\gamma (M,\mathcal {G})} \lambda ^\gamma \frac{1}{1 + |z|^N}\).
The second summand is bounded as
Hence
for \(N > d + \gamma + |\inf _{a \in A} a|\). \(\square \)
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 \((M \times \mathbb {R}) \oplus T^* M\). For readability introduce the symbol \(\varvec{1}\) and decree that it forms a basis for \(\mathbb {R}\). Define the graded vector bundle
with grading \(A(\mathcal {T}) = \{0, 1\}\). For \(q \in M\) let \(\mathcal {T}|_q = {\text {span}}\{ \varvec{1} \} \oplus T^*_q M\) be the fiber at q. A generic element of \(\mathcal {T}_q\) will be written as
with \(a \in \mathbb {R}, \omega \in T^*_q M\). Let \(\mathcal {U}_q := B_{M}(q, \delta )\), where \(\delta \) is the radius of injectivity of M. Define the linear map \(\Pi _q: \mathcal {T}_q \rightarrow D'(\mathcal {U}_q)\) as
Note that, since \(\mathbb {R}\varvec{1}\) is a trivial fiber bundle, it is enough to specify it on the basis element \(\varvec{1}\). This is not possible on \(T^* M\). Note also that \(\Pi _q \omega \) is chosen to have value 0 and differential \(\omega \) at q.
Finally define the re-expansion maps \(\Gamma _{p \leftarrow q}: \mathcal {T}_q \rightarrow \mathcal {T}_p\) as
which is well-defined for \(d(p,q) < \delta \). \(\Pi \) and \(\Gamma \) together form the polynomial model, where we take \(\delta _M = \delta \) in Definition 14.
Remark 26
Note that in Euclidean space with \(\omega = dx_i\) we have
the classical linear polynomials “based” at q. Moreover
so we recover Hairer’s definition [9].
The transport of \(\omega \in T_q^*M\) is chosen such that \(\Pi _q \omega \) and \(\Pi _p \Gamma _{p\leftarrow q} \omega \) have, at p, the same value and the same first derivative. Our re-expansion is not exact, i.e. we do not have \(\Pi _q \tau = \Pi _p \Gamma _{p \leftarrow q} \tau \), but we have the following.
Lemma 27
For \(\omega \in T^*_q M\), uniformly for d(p, q) bounded, \(\ell =0,1,2\), and V differential operators of order \(\ell \)
Proof
Let
By construction \(f(p) = g(p), d|_p f = d|_q g\) and hence the statement follows from Taylor’s theorem. \(\square \)
Remark 28
In the setting of the previous Lemma, not only \(f(p) = g(p)\) but also \(f(q) = g(q)\). Indeed, for two points \(p,q\in M,\) at distance smaller than the cut locus and \(\omega _q\in T^*_q M\),
where the tangent map satisfies indeed \(d|_p (\exp _q^{-1}):T_pM\rightarrow T_q M.\) By definition,
does a priori disagree with \(\Pi _q(\omega _q)=\omega _q\circ \exp _q^{-1}, \) but at p. Let us set \(v_q= \exp _q^{-1}(p)\) and \(v_p=\exp _p^{-1}(q).\) The path \(\gamma =(\exp _q((1-t) v_q))_{0\le t\le 1}\) is the unique path from p to q, with length and speed both equal to d(p, q), staying within the cut-locus from y, that is \((\exp _p( t v_p ))_{0\le t\le 1}:\) in other words, for any \(0\le t\le 1,\)
Hence,
and
The next lemma follows from Lemma 27 and is shown in more generality in Theorem 91.
Lemma 29
The above is a model of transport precision \(\beta =2\).
As a sanity check for our construction, we mention the following lemma, which is almost immediate in the flat case (see [9, Lemma 2.12]). We will prove it in Sect. 9 in a more general setting.
Lemma 30
For \(\gamma \in (1,2)\), a function \(f: M \rightarrow \mathbb {R}\) is in \(C^\gamma (M)\) if and only if there exists a section \(\hat{f}(p) = f_0(p) \varvec{1} + f_1(p) \in \mathscr {D}^\gamma (M,\mathcal {T})\) with \(f_0(p) = f(p)\) and \(f_1(p) \in T^*_p M\).
In that case: \(f_1(p) = d|_p f\).
5 The regularity structure for PAM on a manifold
In the next five sections M is a 2-dimensional closed manifold.
The regularity structure for PAM will be built on two copies of the vector bundle, \(\left( M\times \mathbb {R}^{2}\right) \oplus T^{*}M.\) We denote these two copies by \(\mathcal {V}\) and \(\mathcal {W}\). In order to distinguish the different elements of these bundles we introduce the symbols \(\left\{ \varvec{\mathbf {1}},\varvec{\Xi } ,\mathcal {I}[\varvec{\Xi }],\mathcal {I}[\varvec{\Xi }]\varvec{\Xi }\right\} \) and decree that they form a basis for \(\mathbb {R}^{4}.\) We then write
where \(\varvec{\Xi }T^{*}M\) is simply another copy of \(T^{*}M.\) Formally we have, \(\mathcal {V}=\mathcal {W}\varvec{\Xi }\). As before we will let \(\mathcal {T}|_{p},\mathcal {V}|_{p},\) and \(\mathcal {W}|_{p}\) denote the fibers of these bundles over \(p\in M.\)
The vector bundles \(\mathcal {V}\) and \(\mathcal {W}\) are graded, with gradings
for some \(\alpha \in (-3/2,-1)\) corresponding to the regularity of the driving white noise \(\xi \).
For \(\beta \in A(\mathcal {V})\) (or \(\beta \in A(\mathcal {W}))\) recall (Definition 13) that \({\text {proj}}_{\beta }:\mathcal {V}\rightarrow \mathcal {V}\) (\({\text {proj}}_{\beta }:\mathcal {W}\rightarrow \mathcal {W})\) is the projection taking an element to its \(\beta \) – component. To be concrete, generic elements \(\tau \in \mathcal {V}|_{p}, \tau ' \in \mathcal {W}|_p\) are of the form
with \(a,b,d,e\in \mathbb {R},c,f\in T_{p}^{*}M\). 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 \(\beta \in A\), \(\tau \in \mathcal {V}|_{p}\) (or \(\tau \in \mathcal {W}|_p\)) we write as before, in a slight abuse of notation, \(||\tau ||_{\beta }:=||{\text {proj}}_{\beta }\tau ||\).
The model we shall use for the parabolic Anderson model will be time dependent, so we need slight extensions of our definitions.
Definition 31
For \(\mathcal {G}= \mathcal {V}, \mathcal {W}\), assume we are given a family of models \((\Pi ^t, \Gamma ^t)\) on M parametrized by \(t \in [0,T]\). Define
where \(||\Pi ^t,\Gamma ^t||_{\beta ,M}\) is defined in Definition 14. Note that for fixed t, the model comes with a reconstruction operator (Theorem 23), which we shall denote \(\mathcal {R}_t\).
Definition 32
(Time-dependent modelled distributions) For \(\mathcal {G}= \mathcal {V}, \mathcal {W}\), given a family of models \((\Pi ^t, \Gamma ^t)\) parametrized by \(t \in [0,T]\), denote by \(\mathscr {D}^{t,\gamma }(\mathcal {G}) = \mathscr {D}^{t,\gamma }(M,\mathcal {G})\) the corresponding spaces of modelled distributions. That is, as defined in Definition 18,
For \(\mathfrak {N}> 0\), define the modified norm
Here \(\mu = \alpha ,\) if \(\mathcal {G}= \mathcal {V}\) and \(\mu = 0,\) if \(\mathcal {G}= \mathcal {W}\).
Define \(\mathscr {D}^{\gamma ,\gamma _0}_T(M,\mathcal {G})\) to be the space of functions \(f: [0,T] \times M \rightarrow \mathcal {G}\) with \(f(t) \in \mathscr {D}^{t,\gamma }(M,\mathcal {G})\) and
where \(\upsilon = \alpha ,\) if \(\mathcal {G}= \mathcal {V}\) and \(\upsilon = 0,\) if \(\mathcal {G}= \mathcal {W}\). For \(\mathfrak {N}> 0\), define the modified norm
Remark 33
The modified norms with scaling parameter \(\mathfrak {N}\) are necessary for the fixpoint argument, see Remark 39. As usual with Hölder-type spaces on compact domains, these spaces are complete metric spaces.
Remark 34
Note that we do not need transport in time, as opposed to the definition in [11, Definition 2.4].
The price we pay, is that Hölder regularity in time of the reconstruction of a solution is established in a roundabout way. Namely, by first verifying time regularity of the 0 component of the solution (Theorem 38) and then checking that reconstruction is given by projection on that component (Lemma 24).
In [11] it follows from the definition of a controlled distribution and their reconstruction theorem, Theorem 2.11.
We now build the model for the structures \(\mathcal {V}, \mathcal {W}\). As input we need realizations of \(\varvec{\Xi }\) and \(\mathcal {I}[\varvec{\Xi }] \varvec{\Xi }\).
Definition 35
Assume for \(T>0\) we are given \(\xi \in C^\alpha (M)\) and a family of distributions \(Z^t_p \in C^\alpha (M)\), \(t \in [0,T], p \in M\), satisfying
where the action of the heat kernel \({\mathsf {p}}\) on \(\xi \) is well-defined by Theorem 37. Define
where \(r := -[\alpha ]\) and \(\delta \) is the radius of injectivity of M.
In our application to white-noise forcing, \(\xi \) will be the white noise on M and Z will be constructed via Gaussian renormalization in Sect. 8.
Now define the models for \(\mathcal {V}\) and \(\mathcal {W}\) as
with transports
Lemma 36
These are in fact models with \(\delta _M = \delta \) the radius of injectivity of M and the distances/norms of the model only depend on \(\xi , Z\). Indeed for \(\mathcal {G}= \mathcal {V}, \mathcal {W}\), \(\gamma \in \mathbb {R}\)
with \(\beta = 2\) for \(\mathcal {G}= \mathcal {W}\) and \(\beta = 2 + \alpha \) for \(\mathcal {G}= \mathcal {V}\).
Proof
Let \(\varphi \in \mathcal {B}_{T_p M}^{2,\delta _M}\). By Lemma 11
By definition
Moreover
since \(\omega _p \exp ^{-1}_p \varphi ^\lambda _p = \lambda \psi ^{\lambda }_p\), with \(\psi (\cdot ) = \omega _p(\cdot ) \phi (\cdot )\).
Regarding transport, both the transport of \(\Xi \) and \(\mathcal {I}[\Xi ] \Xi \) are exact by definition and
where we used Lemma 27 for the last step.
Finally
by the Schauder estimate Theorem 37, and
by Lemma 29. Hence
when \(\beta := 2 + \alpha \).
Analogously, one gets the bounds for \(\mathcal {W}\) with \(\beta = 2\). \(\square \)
6 Schauder estimates
Let \({\mathsf {p}}\) be the heat kernel on M. We start with a proof Schauder estimate for distributions, as a warm-up to the one for modelled distributions.
Theorem 37
Let \(T > 0\), and \(F \in L^\infty ([0,T],C^\alpha (M))\), for \(\alpha \in (-2,-1)\). Then for \(t \in [0,T]\)
Proof
As in the proof of the next theorem we shall focus on the singular part \({\mathsf {p}}^N\) in the decomposition \({\mathsf {p}}= {\mathsf {p}}^N + R^N\) using heat asymptotics, Theorem 43, for arbitrary large N. Let us set \(||F||= \sup _{t\le T} ||F_t||_{C^\alpha (M)},\)\(d= d(p,q)\) and let \(\gamma \) be a geodesic path from p to q of constant speed d. We single out the singularity of the heat kernel considering separately the cases \( t \le d^2\) and \(t>d^2\) where we bound the integral over \((0, t-d^2)\) and \((t-d^2,t).\) Close to the singularity, using the first item of Lemma 44,
for \(t>d^2,\) whereas for \(t\le d^2,\)
When \(t>d^2,\) we write on the other interval,
where \(\bullet \) stands for the variable that is being differentiated and \(\cdot \) is the variable in the distribution’s pairing. Then, according to the second item of Lemma 44, for any \(u\in (0,1),\)
The latter four inequalities yield the claim for the singular part of the heat kernel. Now, according to Theorem 43, for N large enough,
This concludes the proof. \(\square \)
We now prove an extension of this classical result to the space of modelled distributions. ForFootnote 4
an element of \(\mathcal {D}^{\gamma ,\gamma _0}_T( \mathcal {V})\), define
with
The well-definedness of these terms is part of the following theorem.
Theorem 38
(Schauder estimate) For \(\alpha \in (-4/3,-1),\) with \(\gamma \in (0,2\alpha + 8/3)\), set \(\varepsilon := (2\alpha + 8/3 - \gamma )/4\) and \(\gamma _0 = \alpha /2 + 1 - \varepsilon \). Let \(T > 0\) and \(f \in \mathcal {D}^{\gamma ,\gamma _0}_T( \mathcal {V})\). Then, for all \(t\in [0,T],\)Footnote 5
Moreover, \(\mathcal {K} f \in \mathcal {D}^{\bar{\gamma }, \bar{\gamma }_0}(\mathcal {W})\), with \(\bar{\gamma } = \gamma + 4/3\), \(\bar{\gamma }_0 = \gamma _0\) and
Remark 39
Here we can see why we introduced the modified norm \(||.||_{\mathcal {D}^{\bar{\gamma }, \bar{\gamma }_0, \mathfrak {N}}_T(\mathcal {W})}\). Without it, i.e. with \(\mathfrak {N}\equiv 1\), the factor on the right hand side cannot be made small, which is necessary for the fixpoint argument.
Remark 40
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 \(\gamma , \gamma _0\) 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.Footnote 6 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 Sect. 9. 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 41
The following proof based on the heat kernel (almost) being a scaled test function goes back, in the flat case, to [4]. A proof splitting up the heat kernel into a sum of smooth, compactly supported kernels (following the strategy of [9]) is also possible, but less convenient.
Proof of Theorem 38
Once the second statement is established, the first one 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 24.
Recall that \(\delta _M = \delta \), the radius of injectivity. By Remark 19 we can, and will only consider points at distance less than \(\delta /4\).
Introduce the short notation
Note that \(||\xi ||_{C^\alpha (M)} \le C_\Pi \).
We shall need the following facts. Since
we have
where we used the classical Schauder estimate Theorem 37.
Moreover for a function \(\varphi \) satisfying the assumptions of Lemmas 17 and 25
and similarly
With these estimates at hand, we shall now control each term in the definition of the norm \(||\mathcal {K} f||_{\mathcal {D}^{\bar{\gamma }, \bar{\gamma }_0, \mathfrak {N}}_T(\mathcal {W})}\), using the decomposition of the heat kernel \({\mathsf {p}}= {\mathsf {p}}^N + R^N\), from Theorem 43, for N large enough, as we did above for the classical Schauder estimate.
Space regularity
Homogeneity 0
This term can be written as
Regarding the contribution of the regular part \(R^N\) of the heat kernel, we write
where \(\nabla \) acts on the dummy variable \(\bullet \) and convolution acts on \(\cdot \) and \(\gamma \) 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 \({\mathsf {p}}^N\). Denoting by g(t, s) the integrand of the above integral, for \(s \in [t-d(p,q)^2,t]\),
The first term we bound as
where we used (10) together with Lemma 44 (i), as well as the Hölder continuity of \(f_\alpha \) in space (9) and in time.
The second we bound as
where we used (10) together with Lemma 44 (i) as well as the Hölder continuity of \(f_\alpha \) in time.
The last one we bound as
where we used (10) together with Lemma 44 (ii) as well as the Hölder continuity of \(f_\alpha \) in space (9) and in time.
Hence
and then by Lemma 42
if
Then the following are upper bounds to \(\bar{\gamma }\)
Both are satisfied under our assumptions.
Now consider \(s \in [0,t-d(p,q)^2]\). By Theorem 62 we have
where \(\gamma (r) := \exp _q( r v ), v := \exp _q^{-1}(p),\) for any \(r\in [0,1],\) and \(\nabla ^2\) is acting on the first variable of \({\mathsf {p}}^N\). Now
The first term we bound as
where we used (10) together with Lemma 44.Footnote 7
The second term we bound as
where we used Lemma 44 and the Hölder continuity of \(f_\alpha \) in time.
Hence by Lemma 42
if
Then the following are upper bounds to \(\bar{\gamma }\)
Both are satisfied under our assumptions. Hence
Homogeneity\(\alpha + 2\)
so we need
which is satisfied under our assumptions.
Homogeneity 1
We only treat here the terms involving \({\mathsf {p}}^N\) within
We write it as \( \int _0^t g(t,s) ds,\) with
It is enough to bound this expression acting on \(X_p \in T_p M\). Write
For \(s \in [t-d(p,q)^2, t]\) we bound (\(\bullet \) denotes the dummy variable on which \(X_p\) is acting, \(\cdot \) denotes the dummy variable in the distribution-pairing)
where we used (10) together with Lemma 44 (ii), as well as the Hölder continuity of \(f_\alpha \) in time.
Now
where we used (10) together with Lemma 44 (ii) with \(Y_p := d|_p \exp ^{-1}_q( z ) \Bigl (X_p\Bigr )\), as well as the Hölder continuity of \(f_\alpha \) in time.
Hence by Lemma 42
if
Then the following are upper bounds to \(\bar{\gamma } - 1\)
Both are satisfied under our assumptions.
Consider now \(s \in [0,t-d(p,q)^2]\). Again it is enough to bound the term acting on some \(X_p \in T_p M\). For notational simplicity let \(v(z) := d|_z {\mathsf {p}}^N_{t-s}(z,\cdot ) \ d|_p \exp ^{-1}_z \langle X_p \rangle \) and \(\zeta ^s_p := \mathcal {R}_s f(s) - f_\alpha (s,p) \xi \). We then write the term to bound as
Now with \(\gamma (t) := \exp _q( t v ), v := \exp ^{-1}_q(p)\),
where we used (10) together with Lemma 44 (iii).
Similarly
where we used Lemma 44 (iii) and the Hölder continuity of \(f_\alpha \) in time.
Finally
where we used Lemma 44 (ii) and the Hölder continuity of \(f_\alpha \) in space (9).
Hence by Lemma 42
if
Then the following are upper bounds for \(\bar{\gamma } - 1\)
Both are satisfied under our assumptions.
Then
Time regularity
Our definition requires only to bound the time increment in homogeneity 0:
Let us consider first the regular part of the heat kernel. According to Theorem 43, for \(N\ge 4\),
Together with the bound \(\sup _{ 0\le r\le T} \Vert \mathcal {R}_r f(r)\Vert _{C^\alpha }<\infty ,\) using a partition of unity, since \(\alpha >-2,\) it yields
Up to a multiplicative constant, we can now bound the contribution of \(R^N\) by
We now treat the term involving \({\mathsf {p}}^N\). Using (11) and Lemma 44 (i)
Further, again using (11) and Lemma 44 (i)
if
We then need
Both are satisfied under our assumptions.
Then
\(\square \)
We used the following results.
Lemma 42
Let \(\rho _1, \rho _2 \in \mathbb {R}\), \(g: \mathbb {R}^2 \rightarrow [0,\infty )\) and assume for \(A \le t \le T\),
Let \(\rho _1 > -1\) and \(\varepsilon \ge 0\) such that \(\rho _2 - \varepsilon < -1\). Then
Proof
Indeed
and
\(\square \)
The following result on heat kernel asymptotics is classical and its proof can be found for example in [1, Theorem 2.30]; see also [15, Section 3.2]. In these references the norm \(||\cdot ||_{C^\ell (M \times M)}\) 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 43
Let M be a d-dimensional, closed Riemannian manifold and \({\mathsf {p}}\) be the heat kernel on M. Then there exist smooth functions \((\Phi _i(p,q))_{i\ge 0}\) with \(\Phi _i(p,q) = 0,\) for \(d(p,q) \ge \delta /4\), such that if we define for \(N \ge 1\)
we have
Moreover for all \(p \in M\)
Lemma 44
Let
\(\psi , \Phi _i\) smooth and with \(\Phi _i(p,q) = 0,\) for \(d(p,q) \ge \delta /4\).
Let \(p \in M\) and define for z in the range of \(\exp _p^{-1}\), \(Y_p \in T_p M\) a tangent vector and \(Z \in \Gamma (TM)\) a vector field
(Note that because of the small support of \({\mathsf {p}}^N\), these are globally well-defined smooth functions by continuation with zero outside of the range of \(\exp _p^{-1}\).)
Then for any multiindex k, any \(n \ge 0\) and \(\ell = 0,1\).
-
(i)
\(|\partial _t^\ell D^k \varphi _t(z)| \lesssim _{\ell ,n,k} \left( \sqrt{t}\right) ^{-d - k - 2 \ell } \frac{1}{1 + (|z|/\sqrt{t})^n },\)
-
(ii)
\(\ \ |D^k \varphi ^Y_t(z)| \lesssim _{n,k} |Y_p| \left( \sqrt{t}\right) ^{-d - 1 - k} \frac{1}{1 + (|z|/\sqrt{t})^n },\)
-
(iii)
\(|D^k \varphi ^{Y_p,Z}_t(z)| \lesssim _{Z,n,k} |Y_p| \left( \sqrt{t}\right) ^{-d - 2 - k} \frac{1}{1 + (|z|/\sqrt{t})^n }.\)
Proof
The summands of \({\mathsf {p}}^N\) are of the same form, apart from the factors \(t^i\), \(i=0,\dots ,N\). Since for \(i \ge 1\) they improve the singularity at \(t=0\), it is enough to treat \(N=0\).
Then
Since \(z \mapsto \Phi (p,\exp _p(z))\) is smooth, uniformly in p, with support in \(B_{T_p M}(0_p, \delta /4)\) 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 \(z \mapsto \exp (-z^2)\) scaled by a factor of \(\sqrt{t}\), and so part (i) with \(\ell =0\) follows from Remark 9.
Now
The first term is treated as above, now having the additional prefactor \(t^{-1} = \left( \sqrt{t} \right) ^{-2}\).
We write the second term as
where \(\phi (s) := s^2 \exp (-s^2)\) is Schwartz. By Remark 9 part (i) with \(\ell =1\) 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
By Lemma 45
and by Lemma 47
Hence for \(|k-\beta | \le |k| - 1\)
For \(|k-\beta | = |k|\) we have \(|\beta | = 0\) and then
The second statement then follows, since \(s \mapsto s^j \exp (-s^2)\) is a Schwartz function, for any \(j \ge 0\).
The third statement follows in a similar fashion from Lemmas 45 and 46. \(\square \)
Lemma 45
Let \(Y_p \in T_p M\) act on the first component of \(d^2\) as follows
Then
Proof
Since \((p,q) \mapsto d^2(p,q)\) is smooth, we only need to show \(g(z) \lesssim |z| |Y_p|\).
Let \(h(q) = Y_p\left[ d^2(p,q) \right] \).
Fix q and take coordinates \(\exp _q^{-1}\). Then
Then \(|g(z)| = |h(\exp _p(z))| \lesssim |Y_p| |z|\) as desired. \(\square \)
Lemma 46
For \(Y_p \in T_p M\) and a vector field Z let
Then, for any multi-index \(\beta ,\)
Proof
This follows from the fact that \((p,q) \mapsto d^2(p,q)\) is smooth. \(\square \)
Lemma 47
For any multiindex k
Proof
This can be verified using the Faa di Bruno formula. \(\square \)
7 Fixpoint argument
The following lemma follows from a direct application of the definition of modelled distributions.
Lemma 48
Define “multiplication by \(\varvec{\Xi }\)” as the vector bundle morphism \(m^{\varvec{\Xi }}: \mathcal {W}\rightarrow \mathcal {V}\) satisfying
If \(f \in \mathcal {D}^{\gamma ,\gamma _0}_T(M,\mathcal {W})\) then \(m( f ) \in \mathcal {D}^{\gamma ,\gamma _0}(M,\mathcal {V})\) and for \(\mathfrak {N}> 0\)
Theorem 49
Let \(u_0 \in C^\infty (\mathbb {R}^2)\). Define \(v_t := P_t u_0\) and lift it to the regularity structure as
Let \((\xi ,Z)\) be given as in Definition 35 and let \(\Pi ^{t,\mathcal {G}}_p, \Gamma ^{t,\mathcal {G}}_{p\leftarrow q}\) be the corresponding models given by Lemma 36, \(\mathcal {G}= \mathcal {V}, \mathcal {W}\). Let \(\alpha \in (-4/3,-1)\), \(\gamma _0 := \alpha /2 + 1\) and \(\gamma \in (4/3, 2\alpha + 4)\). Then there exists \(T > 0\) and a unique \(u \in \mathscr {D}_T^{\gamma ,\gamma _0}(M, \mathcal {W})\) such that on [0, T]
Proof
We follow a standard fixpoint argument. Denote
Denote for \(f \in B(R,\mathfrak {N})\)
Claim: for any \(\mathfrak {N}> 0,\) there is \(R > 0\) such that \(\Phi ( B(R,\mathfrak {N}) ) \subset B(R,\mathfrak {N})\).
Indeed, by Theorem 38 and Lemma 48, for a constant \(c > 0\) possibly changing from line to line,
since \(\alpha > - 4/3\). Hence for T small enough and \(\mathfrak {N}\) large enough, \(\Phi ( B(R,\mathfrak {N}) ) \subset B(R,\mathfrak {N})\), for any \(R > 0\).
Let us show that \(\Phi \) is a contraction on \(B(R,\mathfrak {N})\): for any \(f,f'\in B(R,\mathfrak {N}),\)
Hence for T small enough and \(\mathfrak {N}\) large enough, \(\Phi \) is a contraction on \(B(R,\mathfrak {N})\) for any \(R > 0\). We therefore get unique existence of a solution for small \(T > 0\). \(\square \)
To apply this theorem to white noise forcing, the only ingredient missing is “lifting” it to a model (Definition 35). This is done in the next section.
8 The Gaussian model
Let \(\xi \) be a white noise on M. We recall that \(\xi \) is a Gaussian process associated to the Hilbert space \(L^2(M, {\text {vol}}_M ),\) on a probability space \((\Omega ,\mathcal {B},\mathbb {P}).\)
Lemma 50
There exists a realization of \(\xi \) such that almost surely for any \(\alpha <-1,\)\(\xi \in C^\alpha (M) \).
Proof
For any coordinate chart \(\psi \) defined on an open subset \(\mathcal {U}\subset M,\) and \(\rho \) a positive function with support in \(\mathcal {U},\)\(\xi _\mathcal {U}=\rho \circ \psi ^{-1} \psi _*\xi \) is a Gaussian process associated to the Hilbert space \(L^2(\mathbb {R}^2, \rho ^2\circ \psi ^{-1} \det (g\circ \psi ^{-1} ) ).\) Note that \(\xi _\mathcal {U}\) has the same law as \(\eta \nu \), with \(\eta := \rho \circ \psi ^{-1} \sqrt{ g \circ \psi ^{-1} }\) and \(\nu \) a white-noise on \(\mathbb {R}^d\). According to [9, Lemma 10.2] \(\nu \) has a version which is almost surely in \(C^\alpha (\mathbb {R}^d)\) and hence \(\xi _\mathcal {U}\in C^\alpha (\mathbb {R}^d)\).
Let now \((\rho _i)_{1\le i\le n}\) be a partition of unity subordinated to an atlas \((\mathcal {U}_i,\psi _i)_{1\le i\le n}.\) Then, there is a realization of \((\xi _{\mathcal {U}_i})_{1\le i\le n}\) such that almost surely for all \(\alpha <-1, i\in \{1,\ldots , n\},\)\(\xi _{\mathcal {U}_i}\in C^\alpha (\mathbb {R}^2).\) Then, \(\sum _{i=1}^n \psi _i^*\xi _{\mathcal {U}_i}\) is a realization of \(\xi \) belonging almost surely to \(C^\alpha (M)\). \(\square \)
Thanks to this realization, we can already define the transport map used in the following Lemma (point (i)).
Lemma 51
Let \(\xi \) be the white noise on M and \(Z^t_p\), \(p \in M, t \in [0,T]\) be a collection of random distributions on M such that for some \(\alpha \in (-4/3,-1)\), some \(\kappa ,\delta > 0\),
-
(i)
\(Z^t_q(z) = Z^t_p(z) + \int _0^t \Big \langle {\mathsf {p}}_{t-r}(p,\cdot ) - {\mathsf {p}}_{t-r}(q,\cdot ), \xi \Big \rangle dr\ \xi (z)\),
-
(ii)
$$\begin{aligned} \sup _{p\in M, 0\le t,s\le T, \lambda \in (0,1],\varphi \in \mathcal {B}_{T_p M}^{-[\alpha ],\delta }} \lambda ^{-2(2\alpha +2)-\kappa }\mathbb {E}[ \langle {Z^0_p,\varphi ^\lambda _p}\rangle ^2+|t-s|^{-\kappa }\langle {Z^t_p-Z^s_p,\varphi ^\lambda _p}\rangle ^2] < \infty , \end{aligned}$$(12)
-
(iii)
for any \(\varphi \in C^\infty (M),t \in [0,T], p \in M\): \(\langle {Z^t_p,\varphi }\rangle \) is in the second Wiener chaos.
Then, there is a version of Z and a constant \(h>0\) such that a.s.
Proof
For \(t>s\ge 0,\) define for a chart \((\Psi ,\mathcal {U})\)
Note that \(\bar{Z}^{s,t}_x, x \in \Psi (\mathcal {U})\) and \(\bar{\xi }\) are elements of \(D'( \Psi (\mathcal {U}) )\). Then
where we denote \(S^{s,t}( x \leftarrow y ) := \langle \int _s^t \left[ {\mathsf {p}}_{t-r}(\Psi ^{-1}(x),\cdot ) - {\mathsf {p}}_{t-r}(\Psi ^{-1}(y),\cdot ) \right] dr, \xi \rangle \).
Define the regularity structure and model (in the stronger sense of [9])
and the sector (in the sense of [9, Definition 2.5])
One can then apply [9, Proposition 3.32] to get for every compactum \(K_- \subset \subset K \subset \Psi (U)\), andFootnote 8\(\varphi \in \mathcal {B}_{\mathbb {R}^d}^{r,1}\), \(r := -[\alpha ]\), with \({\text {supp}}\,\varphi \subset K_-,\)
Then, for \(q\in \mathbb {N}\) large enough and any \(\delta >0,\) using (iii), equivalence of moments and then (i)
Let now \((\Psi _i, \mathcal {U}_i)\) be a finite atlas with subordinate partition of unity \(\phi _i\) and \(\delta >0\) be the radius of injectivity of M. Then for \(s,t \in [0,T]\), \(p \in M\), \(\varphi \in \mathcal {B}_{T_p M}^{r,\delta }\)
Now for \(\lambda \) small enough, \(\phi _i \varphi _p^\lambda \not \equiv 0\) implies that \({\text {supp}}\,\varphi ^\lambda _p \subset \mathcal {U}_i\) and in particular \(p \in \mathcal {U}_i\). Hence
where \(\bar{Z}^{s,t;i}_x := (\Psi _i)_* (Z^t_{\Psi _i^{-1}(x)}-Z^s_{\Psi _i^{-1}(x)})\). We can apply Remark 7 to \(\left( \phi _i \varphi ^\lambda _p \right) \circ \Psi _i^{-1}\) and can estimate, using (15),
here for every i, \(\mathcal {K}_i\) is some compactum satisfying \(\Psi _i({\text {supp}}\,\phi _i) \subset \subset \mathcal {K}_i \subset \Psi _i(\mathcal {U}_i)\). Then, by (16),
Let us formulate a setting where we can apply Kolmogorov’s continuity theorem in time. Endow the linear space \(\chi \) of maps \(Y: M\rightarrow \mathcal {D}'(M)\), such that for any \(p\in M,\)\({\text {supp}}(Y_p)\subset B(p,\delta /2)\), with the norm
and consider the Banach space \(\chi _{2\alpha +2}=\{Y\in \chi : ||Y||<\infty \}.\) Using Remark 7, we apply this to
Here \(\rho _p := \rho \circ \exp ^{-1}_p\), with \(\rho \) smooth, \({\text {supp}}\,\rho \subset B_{T_p M}(0_p, \delta /2 - \epsilon )\) for some \(\epsilon > 0\) small enough.
Then, from the argument before, for any \(s,t\ge 0\) and q large enough, we have
The result now follows from the Kolmogorov continuity theorem. \(\square \)
A simple way to define \(Z^t_p\) is here to recenter the terms involving one product of distributions by their mean; this is an instance of a Wick product, see for instance [8]. For any \(t> 0,\) the heat kernel and the heat operator are denoted respectively by \({\mathsf {p}}_t:M^2\rightarrow \mathbb {R}\) and \(P_t,\) and we write for \(p\in M,\)\(\mathsf {q}_t(p)={\mathsf {p}}_t(p,p).\) According to Lemma 50 and Theorem 43, we can consider \(P_t(\xi )\) as a function and the map \(t\in \mathbb {R}_{> 0} \mapsto P_t(\xi ) \in C^{\infty }(M) \) is continuous.
We set for any \(p\in M, t\in \mathbb {R}_{\ge 0}\) and any function \(\varphi \in C^{\infty }(M),\)
where for any \(s>0\) and \(\varphi \in C^{\infty }(M),\)
Note that for any \(s>0,\)
For any \(t\ge 0,\) let us consider the operator \(K_t= \int _0^t P_s ds\) and for any \(p,q\in M\) with \(p\not =q,\) set \(\mathsf {k}_t(p,q)=\int _0^t {\mathsf {p}}_{s}(p,q)ds .\) Let us note that the operator
has a continuous kernel according to Theorem 43, that we shall denote \(\mathsf {k}_{2,t}\).
Remark 52
Note that, in (17) we subtract a function depending on space. This is different than the renormalization in the flat case, [9, Section 9.1], where just a constant is subtracted.
This could be avoided, by realizing that the only factor contributing to the blowup of
is, by Theorem 43,
which is independent of space. Since we do not need the renormalization to be independent of space (or time, for that matter), we do not pursue this.
Proposition 53
For any \(t\in \mathbb {R}_{\ge 0},\) almost surely for any \( p \in M\) and \(\varphi \in C^{\infty }(M),\)\(\langle Z^t_p,\varphi \rangle \) is well-defined and there exists a modification of the process given by \((\langle Z^t_p,\varphi \rangle )_{p\in M, \varphi \in C^{\infty }(M), t\ge 0 }\) such that almost surely (13) holds true.Footnote 9
Proof of Proposition 53
It is enough to prove the assumption of Lemma 51. Let us fix \(p \in M\) and \(T>0\). Let \(\delta \) be the radius of injectivity of M.
Let us first check that for any \(\varphi \in C^{\infty }(M),\)\(Z^t_x(\varphi ),\) is well defined for \(0\le t\le T.\) Therefor, let us recall – see Theorem 43 – that
The Wick formulas imply for any \(0<s<T,\)
It follows that
Besides, \(\mathbb {E} \left( \langle {\xi , \varphi }\rangle ^2 P_s\xi (p)^2\right) = q_{2s}(p)\Vert \varphi \Vert _2^2+ P_s(\varphi )(p)^2\le s (L+1) \Vert \varphi \Vert _\infty ^2,\) so that \(Z^t_p(\varphi )\) is well defined. We shall now prove that for any \(\kappa <0, \)
which together with Lemma 50 shall yield the claim. We fix now \(\kappa <0.\) Let us first prove that the expectation of the second integrand in \(\Pi _p^t(\mathcal {I}(\Xi )\Xi )\) is almost surely of homogeneity \(\kappa \). Indeed, according to Theorem 43, there exists \(C_T>0\), such that, for all \(0<t < T, p,q\in M,\) with \(p\not = q,\)
Since
it follows that for any \(\kappa <0,\)
Setting \(I_{p,s}= \xi P_s \xi - \xi P_s \xi (p)\) and \(:I_{p,s}:\ = I_{p,s} -\mathbb {E}[I_{p,s}],\)Footnote 10 it remains to estimate
For any \(\varphi \in \mathcal {B}_{T_p M}^{2,\delta }, s,s' \ge 0,\)
and
where the second line follows from the Cauchy-Schwarz inequality. It follows from Lemma 54 below, that there exists \(C>0\), such that for any \(\varphi \in \mathcal {B}_{T_p M}^{r,\delta }, t \in [0,T],\)
Hence for any \(\lambda \in (0,1]\) and \(\varphi \in \mathcal {B}_{T_p M}^{2,\delta }\),
\(\square \)
Lemma 54
For any \(\nu>\eta >0,T\ge 0\), there exists \(C>0,\) such that for any \(q\in M, t\in [0,T],\)
Proof
On the one hand, according to (18) and Theorem 43, the left-hand-side of (22) is uniformly bounded by \(C_T t,\) for all \(t\in [0,T],\) for some \(C_T>0.\) On the other hand, the estimate (22) would hold true, with \(\eta =0,\) if \(K_t\) would be replaced by a \(C^2\) symmetric function on \(M^2\). Indeed if \(K:M^2\rightarrow \mathbb {R}\) is a \(C^2\) symmetric function,
where the index below the connexion symbol indicates the variable on which the latter is acting, and \(\gamma \) is a geodesic from p to q. According to Theorem 43, one can therefore consider \(K^{2,N}_{t}= \int _0^t s P_s^N ds \) in place of \(K^2_t,\) as soon as N is large enough. This same theorem ensures that there exists a smooth function \(\Phi :[0,T]\times M^2\rightarrow \mathbb {R}_{\ge 0}\) such that for all \(\tau \in (0,T], p,q\in M,\)
Let us set \(q_\tau (r)= \frac{1}{2\pi \tau } e^{-\frac{r^2}{\tau }},\) for any \(r,\tau >0.\) We shall apply (23) to \(K_{t,\varepsilon }= \int _{\varepsilon }^t sP^N_s ds,\) for any fixed \(\varepsilon >0.\) Up to a constant, the integrand of the right-hand-side of (23) is bounded by \( \int _\varepsilon ^t\left( d(p,q)\Vert \nabla _{1,\dot{\gamma }_s} q_\tau \circ d (\gamma _s,\gamma _r) \Vert +\Vert \nabla _{1,\dot{\gamma }_s} \nabla _{2,\dot{\gamma }_r} q_\tau \circ d (\gamma _s,\gamma _r)\Vert \right) d\tau . \) Let us set \(R=d(p,q).\) The first term can be bounded by
and the second by
for some constant \(C_T>0.\) These two bounds, once integrated in (23), imply that for any \(\alpha >0,\) the left-hand-side of (22) is bounded by \(C_T d(p,q)^{2-\alpha },\) uniformly on \(p,q\in M\) and \(t\in [0,T].\) Using the bound \(\min \{a,b\}\le a^\eta b^{1-\eta },\) for \(a,b,\eta \in (0,1),\) gives (22). \(\square \)
9 Appendix - Higher order “polynomials”
We recall the regularity structure of polynomial functions in flat space \(\mathbb {R}^{d}\) given in [9]. It is used to abstractly describe functions in \(C^{\gamma }\left( \mathbb {R}^{d}\right) \), \(\gamma >0\), and also forms a central ingredient for general regularity structures associated with singular SPDEs. Let \(\gamma >0\) and \(n=\lfloor \gamma \rfloor \), that is \(n\in \mathbb {N}_{0}\) and \(\gamma \in (n,n+1]\). For simplicity of notation let \(d=1\). Define
where \({\text {span}}\{X^{\ell }\}\) denotes the one-dimensional vector space spanned by the abstract symbol \(X^{\ell }\). Hence \(\mathcal {T}_{flat} \simeq \mathbb {R}^{n+1}\).
Given \(x,y\in \mathbb {R}\) and \(\ell \in \mathbb {N}_{0},\) we define the linear maps, \(\Pi _{x}:\mathcal {T}^{flat}\rightarrow \subset \mathcal {D}^{\prime }\left( \mathbb {R}\right) \) and \(\Gamma _{x\leftarrow y}:\mathcal {T}^{flat}\rightarrow \mathcal {T}^{flat},\) which are uniquely determined by
In this case one has \(\Pi ^{flat}_{x}\Gamma ^{flat}_{x\leftarrow y}\tau =\Pi ^{flat}_{y}\tau \) for all \(\tau \in \mathcal {T}_{flat}.\) One can use this regularity structure to describe regular functions.
Lemma 55
([9, Lemma 2.12]) Let \(f:\mathbb {R}\rightarrow \mathbb {R}\). Then \(f\in C^{\gamma }(\mathbb {R})\) if and only if there exists \(\hat{f}:\mathbb {R}\rightarrow \mathcal {T}^{flat}\) with \(\hat{f}_{0}(x)=f(x)\) and
In that case \(\hat{f}_{\ell }(x)=f^{(\ell )}(x),\ell =0,\dots ,n\).Footnote 11
9.1 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 closed d dimensional Riemannian manifold. In order to do to this we need to store higher order derivatives of functions \(f:M\rightarrow \mathbb {R}\) 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 \(f:M\rightarrow \mathbb {R}\) on a Riemannian manifold with Levi–CivitaFootnote 12 connection \(\nabla \) (see for example [14, Lemma 4.6]).
Definition 56
Define \(\nabla ^{\ell }|_{p}f\in \left[ T_{p}^{*}M\right] ^{\otimes \ell }\cong \left[ T_{p}M^{\otimes \ell }\right] ^{*}\) by,
and then inductively by;
where \(X_{1},\dots ,X_{\ell }\) are arbitrary vector fields on M.
A few remarks are in order.
-
1.
As the notation suggests, \(\nabla ^{\ell }|_{p}f\) is indeed tensorial, i.e. the right side of the previously displayed equation really only depends on the vector fields, \(\left\{ X_{i}\right\} _{i=1}^{\ell }\), through their values at p.
-
2.
In the literature \(\nabla f\) sometimes denotes the gradient of f. We never use the gradient of a function in this work.
-
3.
We shall also sometimes write \(\nabla _{W}^{\ell }f=\langle \nabla ^{\ell }|_{p}f,W\rangle \) for any \(W\in (T_{p}M)^{\otimes \ell }.\)
For a curve \(\gamma \) in M we will write
for parallel translation along \(\gamma \). For a tensor field W along \(\gamma \) we write
for the covariant derivative of W along \(\gamma \).
Lemma 57
If f is an \(\ell \)-times continuously differentiable function in a neighborhood of \(p\in M,\)\(v\in T_{p}M,\) and \(\gamma _{v}(t):=\exp _{p}(tv),\) then
More generally, if \(\ell ,n\in \mathbb {N}_{0},\)f is an \((\ell +n+1)\)-times continuously differentiable function in a neighborhood of \(p\in M\) and \(W_{t}:=//_{t}(\gamma _{v})^{\otimes \ell }W_{0},\) then
Proof
Let \(\gamma _{v}\left( t\right) :=\exp _{p}\left( tv\right) \) so that \(\gamma _{v}\left( t\right) \) solves the geodesic differential equation, \(\nabla \dot{\gamma }_{v}\left( t\right) /dt=0\) with \(\dot{\gamma }_{v}\left( 0\right) =v.\) The proof is completed by showing (by induction) that
The case \(k=1\) amounts to the definition that \(\nabla _{v}f=vf=df\left( v\right) \) for all \(v\in TM.\) For the induction step we have by the product rule;
wherein the last equality we have again used the product rule to conclude that \(\frac{\nabla }{dt}\left[ \dot{\gamma }_{v}\left( t\right) ^{\otimes k}\right] =0.\) The result now follows by evaluating (27) at \(k=\ell \) and \(t=0.\) The more general assertion in (26) is proved similarly. One only need to observe that \(\frac{\nabla }{dt} W_{t}=0\), by definition of parallel transportation, and hence the presence of \(W_{t}\) in the expressions in no way changes the computations. \(\square \)
Definition 58
(Symmetrizations) If V is a real vector space and \(\ell \in \mathbb {N},\) we let \({\text {Sym}}_{\ell }:V^{\otimes \ell }\rightarrow V^{\otimes \ell }\) denote the symmetrization projection uniquely determined by
where \(S_{\ell }\) is the permutation group on \(\left\{ 1,2,\dots ,\ell \right\} .\) Often we will simply write \({\text {Sym}}\) for \({\text {Sym}} _{\ell }\) as it will typically be clear what \(\ell \) 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 \(\left\langle \cdot ,\cdot \right\rangle \) denote the pairing between a vector space and its dual. We will often identify \(\left( V^{*}\right) ^{\otimes \ell }\) with \(\left[ V^{\otimes \ell }\right] ^{*}\) where the identification is uniquely determined by
We also identify \(\left( V^{*}\right) ^{\otimes \ell }\) with the space of multi-linear maps from \(V^{\ell }\rightarrow \mathbb {R}\) using,
Under these identification we have
Remark 59
If \(T\in \left[ V^{*}\right] ^{\otimes \ell }\) and \(v_{1} ,\dots ,v_{\ell }\in V,\) then
and therefore,
This formula shows that the symmetric part \({\text {Sym}}\left[ T\right] \) of T is completely determined by the knowledge of \(T\left( v,v,\dots ,v\right) \) for all \(v\in V.\)
Definition 60
Let \(\Sigma ^{\ell }T_{p}^{*}M\) denote the symmetric tensors in \(\left[ T_{p}^{*}M\right] ^{\otimes \ell }\) and for \(T\in \left[ T_{p}^{*}M\right] ^{\otimes \ell },\) let \({\text {Sym}}[T]\in \Sigma ^{\ell }T_{p} ^{*}M\) denote the symmetrization of T as above.
Example 61
If U is an open subset of M and f is \(\ell \)-times continuously differentiable on U, then \({\text {Sym}}\left[ \nabla ^{\ell }f\right] \) defines a local section (over U) of \(\Sigma ^{\ell }T^{*}M.\) Moreover since \(v^{\otimes \ell }\) is symmetric for all \(v\in T_{p}M\) we may write (25) as
Theorem 62
(Taylor’s Theorem on M) Let \(\ell ,n\in \mathbb {N}_{0},\)\(p\in M,\)\(v\in T_{p}M,\)\(\gamma _{v}(t):=\exp _{p}(tv),\)\(//_{t}(\gamma _{v}):T_{\gamma _{v}(0)}M\rightarrow T_{\gamma _{v}(t)}M,\)\(W_{0}\in T_{p}M^{\otimes \ell },\) and \(W_{t}:=//_{t}(\gamma _{v})^{\otimes \ell }W_{0}.\) If f is \((\ell +n+1)\)-times continuously differentiable on U, where U is an open set containing \(\gamma _{v}\left( \left[ 0,1\right] \right) ,\) then
When \(\ell =0\) the previous equation reads as (also see [5, Theorem 6.1])
where
Proof
Let \(g\left( t\right) :=\nabla _{W_{t}}^{\ell }f\) and recall that the standard Taylor’s theorem with remainder states;
The results now follow by using Lemma 57 in order to compute the \(g^{\left( k\right) }\left( t\right) \) for \(1\le k\le n+1.\)\(\square \)
Remark 63
Since parallel translation is isometric it follows (continuing the notation in Theorem 62) that
and hence
Since M is a compact Riemannian manifold it is necessarily complete and therefore, by the Hopf–Rinow theorem, for each \(q\in M\) we may find at least one \(v\in T_{p}M\) such that \(q=\exp _p\left( v\right) \) and \(d\left( q,p\right) =\left| v\right| .\) Using these remarks we can reformulate (30) as follows.
Corollary 64
If f is \((n+1)\)-times continuously differentiable on M, \(p,q\in M,\) and \(v\in T_{p}M\) is chosen so that \(q=\exp _p\left( v\right) \) and \(d\left( q,p\right) =\left| v\right| ,\) then
where
Furthermore if f is n-times continuously differentiable on M then
Definition 65
(Taylor approximations) Suppose that \(U\subset M\) is an open subset of M, \(p\in U,\)f a n-times continuously differentiable function on M and \(\varepsilon >0\) is sufficiently small so that \(B_{M}(p, \varepsilon ) \subset U\) and \(\varepsilon \) is smaller than the injectivity radius of M. We then define, \({\text {Tay}}_{p}^{n}f\in C^{\infty }\left( B_{M}(p, \varepsilon ) \right) \) by
Remark 66
With this notation, Corollary 64 reads as
In the case \(M=\mathbb {R}^{d}\) and f is a polynomial of degree at most n, it follows by Taylor’s theorem that \(f={\text {Tay}} _{p}^{n}f\) for all \(p\in \mathbb {R}^{d}.\) So in the flat case the error term here is no longer present.
Lemma 67
If f is a n-times continuously differentiable function on M and \(f\left( q\right) =o\left( d\left( p,q\right) ^{n}\right) ,\) then \(\left( Vf\right) \left( p\right) =0\) for any \(n^{\text {th}}\) - order differential operator V and in particular, \(\nabla ^{k}|_{p}f=0\) for all \(0\le k\le n.\)
Proof
Let \(\left( \Psi ,U\right) \) be a chart on with \(p\in U\) and \(\Psi \left( p\right) =0\) and define \(F:=f\circ \Psi ^{-1}\in C^{n}\left( \tilde{U} :=\Psi \left( U\right) \right) .\) Then the give assumption implies \(F\left( x\right) =o\left( \left| x\right| ^{n}\right) \) and therefore for any \(x\in \mathbb {R}^{d}\) and \(t\in \mathbb {R}\) small we have \(F\left( tx\right) =o\left( t^{n}\right) \) from which it easily follows that
As \(\left( D^{k}F\right) \left( 0\right) \) is symmetric and \(x\in \mathbb {R}^{d}\) was arbitrary we may conclude that \(\left( D^{k}F\right) \left( 0\right) =\mathbf {0}\in \left( \mathbb {R}^{d}\right) ^{*\otimes k}\) for \(0\le k\le n.\) As any \(n^{\text {th}}\) - order differential operator U on \(C^{n}\left( M\right) \) may be written locally as
for some smooth functions, \(W_{k}:U\rightarrow \left( \mathbb {R}^{d}\right) ^{\otimes k}\) for each \(0\le k\le n,\) it follows that
\(\square \)
Corollary 68
If f a n-times continuously differentiable function on M and V is an \(n^{\text {th}}\) - order differential operator, then
and in particular,
from which it follows that \(\nabla ^{n}|_{p}f\) is a linear combination of \(\left\{ {\text {Sym}}\,\nabla ^{k}|_{p}f\right\} _{k=0}^{n}.\)
We will make the last assertion of Corollary 68 more explicitly in Corollary 74 and Remark 76. The upshot is that there is no loss of information in only keeping track of the symmetrizations of the covariant derivatives.
Corollary 69
If \(f\in C^{\infty }\left( M\right) ,\)\(p,q\in M\) with \(d\left( p,q\right) \) then for \(0\le k\le n\) we have
where \(v:=\exp _{p}^{-1}\left( q\right) .\)
Proof
Let us apply the estimate in (31) with f replaced by \(g:=f-{\text {Tay}}_{p}^{n}f\) keeping in mind that \(\nabla ^{k}|_{p}\left[ f-\left( {\text {Tay}}_{p}^{n}f\right) \right] =0\) for \(0\le k\le n\) by Corollary 68. This allows us to conclude for \(W_{0}\in T_{p}M^{\otimes k}\) that
where \(v:=\exp _{p}^{-1}\left( q\right) .\) As the map \(W_{0}\rightarrow W_{1}\) is an isometry it follows that
\(\square \)
9.1.1 Symmetric parts of covariant derivatives determine all derivatives
We will now make Corollary 68 more precise.
Definition 70
If \(\left( x,U:={\text {dom}}( x ) \right) \) is a chart on M, let \(D^{x}\) denote the flat covariant derivative on TU determined by \(D^{x}\frac{\partial }{\partial x^{j}}=0\) for \(1\le j\le d.\)
Remark 71
If \(V=\sum _{j=1}^{d}V_{j}\frac{\partial }{\partial x^{j}}\) is a vector field on U and \(v\in T_{m}M\) then \(D_{v}^{x}V=\sum _{j=1}^{d}\left( vV_{j}\right) \frac{\partial }{\partial x^{j}}|_{m}.\) Using \(D^{x} \frac{\partial }{\partial x^{j}}=0\), it easily follows that for all \(\ell \in \mathbb {N}\) and any \(\ell \)-times continuously differentiable function f we have
and in particular \(\left( D^{x}\right) ^{\ell }f\in \Sigma ^{\ell }T^{*}U.\)
Lemma 72
Suppose that \(\left( x,U:={\text {dom}}(x) \right) \) is a chart on M, \(D=D^{x}\) is the flat covariant derivative of Definition 70. Then, there exists a family of sections \(Q_{\ell ,n}\in \Gamma \left[ {\text {Hom}}\left[ TU^{\otimes n},TU^{\otimes \ell }\right] \right] \) for \(1\le \ell \le n,\) such that \(Q_{n,n} = {\text {id}}\) and for all n-times continuously differentiable functions f,
Proof
Let \(D=D^{x}\) and \(\Gamma \) be the \({\text {End}}\left( TU\right) \) – valued connection one form on TU so that \(\nabla =D+\Gamma .\) It is enough to verify that (32) holds on a basis for \(T_{p}U^{\otimes n}.\) To this end, let \(i_{j}\in \left\{ 1,2\dots ,d\right\} ,\) for \(1\le j\le n\) and let \(V_{j}=\frac{\partial }{\partial x^{i_{j}}}.\) Then,
which shows that (32) holds for \(n=1.\) For the sake of completing the proof by induction, let us now assume that (32) 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 \(\nabla _{V_{n}\otimes \dots \otimes V_{1}}^{n}f\) may be expressed in the form claimed in (32). \(\square \)
Corollary 73
Let us continue the notation in Lemma 72. Then, there exists
such that \(\bar{Q}_{n,n}={\text {id}}\) and for all n-times continuously differentiable functions f,
Proof
The proof is again by induction on n. For \(n=1,\) we have \(D_{W} f=Wf=\nabla _{W}f,\) so there is nothing to prove. For the inductive step, suppose that (33) holds at level \(n-1\) and below. From (32) with W replaced by \({\text {Sym}}_{n}W,\) it follows that,
wherein the last equality we have used that \(D^{n}f\) is already symmetric. From the previous equation along with the inductive hypothesis, we conclude that \(\left\langle D^{n}f,W\right\rangle \) may be expressed as described in (33). \(\square \)
Corollary 74
If \(\nabla \) is a covariant derivative on TM, then there exists
such that \(Q_{n,n}^{\nabla }={\text {id}}\) and for all n-times continuously differentiable functions f,
Proof
First suppose that \(M=U,\) as in Lemma 72. Then combining the results of Lemma 72 and Corollary 73, there exists \(Q_{\ell ,n}^{x}\in \Gamma \left[ {\text {Hom}}\left[ TU^{\otimes n},TU^{\otimes \ell }\right] \right] \) such that (34) holds for all \(W\in TU^{\otimes M}.\) Let \(\left\{ x_{\alpha }\right\} _{\alpha =1}^{N}\) be a collection of charts on M such that \(\left\{ {\text {dom}}\left( x_{\alpha }\right) \right\} _{\alpha =1}^{N}\) is an open cover of M and \(\left\{ \psi _{\alpha }\right\} _{\alpha =1}^{N}\) be a partition of unity relative to this cover. To complete the proof we define
\(\square \)
We note the following corollary for completeness.
Corollary 75
If \(\nabla \) is a covariant derivative on TM and L is a linear \(n^{\text {th}}\) - order differential operator on \(C^{\infty }\left( M\right) ,\) then there exists smooth sections, \(W_{\ell }\in \Gamma \left( \Sigma ^{\ell }TM\right) \) for \(0\le \ell \le n\) such that
Proof
By definition Lf is locally given by \(Lf=\sum _{\ell =0}^{n}\left\langle D^{n}f,A_{n}\right\rangle \) for some \(A_{n}\in \Gamma \left( \Sigma ^{\ell }TU\right) .\) Using Corollaries 73 and 74, we may locally express Lf as in (35). The global picture may then be constructed using a partition of unity argument. \(\square \)
Remark 76
Our Proof of Corollary 74 was local in nature and hence does not give much information about how the \(Q_{\ell ,n}^{\nabla }\) depend on \(\nabla .\) It is possible to give a global Proof of Corollary 74 which would show that \(Q_{\ell ,n}^{\nabla }\) may be constructed from certain combinations of covariant derivatives of the torsion and curvature tensor of \(\nabla .\) Here is a sketch of this argument. In this sketch we let \(v\wedge w:=v\otimes w-w\otimes v\) for any \(v,w\in T_{p}M.\)
-
1.
If \(v_{1},\dots ,v_{n}\in T_{p}M\) and \(1\le i<n,\) then
$$\begin{aligned}&\nabla _{v_{n}\otimes \dots \otimes v_{i+2}\otimes \left[ v_{i+1}\wedge v_{i}\right] \otimes v_{i-1}\otimes \dots \otimes v_{1}}^{n}f\\&\quad =\left\langle \nabla _{v_{n}\otimes \dots \otimes v_{i+2}}^{n-i-1}\left[ R\left( \cdot ,\cdot \right) \nabla ^{i-1}f\right] ,v_{i+1}\otimes v_{i}\otimes v_{i-1}\otimes \dots \otimes v_{1}\right\rangle \\&\quad \quad +\left\langle \nabla _{v_{n}\otimes \dots \otimes v_{i+2}}^{n-i-1}\left[ \nabla _{T\left( \cdot ,\cdot \right) }\nabla ^{i-1}f\right] ,v_{i+1}\otimes v_{i}\otimes v_{i-1}\otimes \dots \otimes v_{1}\right\rangle \end{aligned}$$where \(R\left( \cdot ,\cdot \right) \nabla ^{i-1}f\) is the appropriate action of the curvature tensor of \(\nabla \) on \(\nabla ^{i-1}f\) and T is the torsion tensor of \(\nabla .\)
-
2.
As a consequence of item 1. and the fact that every permutation is a composition of transpositions, it follows that for any permutation \(\sigma \in S_{n},\)
$$\begin{aligned} \nabla _{v_{\sigma (n)}\otimes \dots \otimes v_{\sigma (1)}}^{n}f=\nabla _{v_{n}\otimes \dots \otimes v_{1}}^{n}f+\sum _{\ell =1}^{n-1}\left\langle \nabla ^{\ell }f,\mathbf {Q}\left( \sigma \right) _{\ell ,n}v_{n}\otimes \dots \otimes v_{1}\right\rangle , \end{aligned}$$(36)where \(\mathbf {Q}\left( \sigma \right) _{\ell ,n}\in \Gamma \left[ {\text {Hom}}\left[ TM^{\otimes n},TM^{\otimes \ell }\right] \right] \) are constructed from certain combinations of covariant derivatives of the torsion and curvature tensor of \(\nabla .\)
-
3.
Summing (36) on \(\sigma \) and then dividing by n! and setting
$$\begin{aligned} \mathbf {Q}_{\ell ,n}=\frac{1}{n!}\sum _{\sigma \in S_{n}}\mathbf {Q}\left( \sigma \right) _{\ell ,n} \end{aligned}$$shows
$$\begin{aligned} \left\langle {\text {Sym}}\,\nabla ^{n}f,v_{n}\otimes \dots \otimes v_{1}\right\rangle =\left\langle \nabla ^{n}f,v_{n}\otimes \dots \otimes v_{1}\right\rangle +\sum _{\ell =1}^{n-1}\left\langle \nabla ^{\ell }f,\mathbf {Q}_{\ell ,n} v_{n}\otimes \dots \otimes v_{1}\right\rangle , \end{aligned}$$(37)where the \(\mathbf {Q}_{\ell ,n}\in \Gamma \left[ {\text {Hom}}\left[ TM^{\otimes n},TM^{\otimes \ell }\right] \right] \) are constructed from certain combinations of covariant derivatives of the torsion and curvature tensor of \(\nabla .\)
-
4.
Using (37) recursively then shows there exists \(Q_{\ell ,n}^{\nabla }\in \Gamma \left[ {\text {Hom}}\left[ TM^{\otimes n},TM^{\otimes \ell }\right] \right] \) such that
$$\begin{aligned} \left\langle \nabla ^{n}f,v_{n}\otimes \dots \otimes v_{1}\right\rangle= & {} \left\langle {\text {Sym}}\,\nabla ^{n}f,v_{n}\otimes \dots \otimes v_{1}\right\rangle \\&+\sum _{\ell =1}^{n-1}\left\langle {\text {Sym}}\,\nabla ^{\ell }f,Q_{\ell ,n}^{\nabla } v_{n}\otimes \dots \otimes v_{1}\right\rangle , \end{aligned}$$where each \(Q_{\ell ,n}^{\nabla }\) is constructed from certain combinations of covariant derivatives of the torsion and curvature tensor of \(\nabla .\)
9.2 The polynomial regularity structure and model
We are now ready to set up to regularity structure for “polynomials” up to order n on a manifold.
Definition 77
Fix \(n\ge 0\) and let \(\mathcal {T=}\bigoplus _{\ell =0}^{n} \Sigma ^{\ell }T^{*}M\) be the vector bundle over M with fiber at \(p\in M\) given by
On each fiber of \(\Sigma ^{\ell }T^{*}M\) we use the norm induced by the Riemannian metric. With this norm, for \(\tau \in \Sigma ^{\ell }T^{*}_p M\), \(X \in (T_p M)^{\otimes \ell }\)
The vector bundle \(\mathcal {T}\) will be used to store higher order derivatives of functions. On flat space \(\mathbb {R}^{d}\) such “abstract Taylor expansions” were realized as honest functions using polynomials, see (24). Polynomials are the simplest function that have specified derivatives at one point. On the manifold we instead choose polynomials in exponential coordinates.
Definition 78
(Realization of an abstract polynomial) For \(\tau =(\tau _{0},\dots ,\tau _{n})\in \mathcal {T}_{p}\) define
These local “Taylor polynomials” are a good substitute for the usual Taylor polynomials in the flat space theory, as Lemma 79 and Corollary 80 below demonstrate.
Lemma 79
Let \(A=A_{0}+A_{1}+\dots +A_{n}\in \mathcal {T}_{p},\) with \(A_{\ell }\in \Sigma ^{\ell }T_{p}^{*}M,\) for \(\ell =0,\ldots ,n,\) and define
Then,
Proof
Let \(\gamma _{v}\left( t\right) =\exp _{p}\left( tv\right) \). Then,
and hence by Lemma 57
which suffices to complete the Proof by Remark 59. \(\square \)
We now have the immediate corollary of this lemma.
Corollary 80
Let \(\tau \in \Sigma ^{\ell }T_{p}^{*}M\). Then, for \(i=0,\dots ,n,\)
Remark 81
Let x be exponential coordinates around \(p\in M\), i.e. suppose that \(x=\left( x^{1},\dots ,x^{d}\right) \) where \(\left\{ x^{i}\left( q\right) \right\} _{i=1}^{d}\) are the coordinates of \(\exp _{p}^{-1}\left( q\right) \) relative to some basis \(\left\{ u_{i}\right\} _{i=1}^{d}\) of \(T_{p}M.\) Then with \(v=\sum _{i=1}^{d}v^{i}u_{i}\in T_{p}M,\)
from which it follows that
Definition 82
(Transportation) Let \(\Gamma _{p\leftarrow q}:\mathcal {T} |_{q}\rightarrow \mathcal {T}|_{p}\) be defined by \(\Gamma _{p\leftarrow q} \tau :=\bar{\tau },\) where
which makes sense for \(d(p,q)<\delta \), the radius of injectivity of M.
Remark 83
For \(n\ge 2\) this transport will in general also go “upwards.” That is, if \(\tau \in \mathcal {T}_{\alpha }|_{y}\) some \(\alpha <n\), then in general \(\Gamma _{x\leftarrow y}\tau \) will have components in homogeneities strictly larger than \(\alpha \). This is not allowed in the original formulation of a regularity structure by Hairer [9, Definition 2.1]. As we have seen in the main text, this poses no problem, since our modified definition of a model (Definition 14) allows for it. We moreover believe that any transport that wants to achieve the following lemma for a “polynomial model” is forced to do this.
The definitions have been arranged so that \(\Pi _{q}\tau \) and \(\Pi _{p} \Gamma _{p\leftarrow q}\tau \) agree at p to order n:
Lemma 84
Let \(\tau \in \mathcal {T}|_{q}\) and \(p,z\in U\) where U is a sufficiently small neighborhood of q. If V is a differential operator of order \(k\le n\) defined on U, then
Proof
Let
so that
where \({\text {Tay}}^n_p\) was defined in Definition 65. Using Corollary 69, we have the estimate,
where \(v:=\exp _{p}^{-1}\left( z\right) .\) For \(d\left( z,p\right) <\varepsilon \) and \(0\le t\le 1,\) let \(\left[ p,z\right] _{t}:=\exp \left( t\exp _{p}^{-1}\left( z\right) \right) \) so that \(t\rightarrow \left[ p,z\right] _{t}\) is the geodesic joining p to z parametrized by \(\left[ 0,1\right] .\) Then we have
and so we have
\(\square \)
For the proof of the first half of Theorem 92 below, it is convenient to introduce as in [5] the notion of a parallelism on a vector bundle, E, over M.
Definition 85
(Diagonal domains) Let \(\mathcal {U}\) be an open set on M. An open set \(\mathsf {D}^{\mathcal {U}}\subseteq M\times M\) is a\(\mathcal {U}\) – diagonal domain if it contains the diagonal of \(\mathcal {U}\), that is \(\Delta ^{\mathcal {U}}:=\bigcup _{p\in \mathcal {U}}\left( p,p\right) \subseteq \mathsf {D}^{\mathcal {U}}\). A local diagonal domain is a \(\mathcal {V}\) – diagonal domain for some nonempty open \(\mathcal {V\subseteq }M\).
If \(\mathcal {U}=M,\) we write \(\mathsf {D}:=\mathsf {D}^{M}\) and refer to \(\mathsf {D}\) simply as a diagonal domain.
Definition 86
(Parallelisms) Let E be a vector bundle over M and \({\text {Hom}}\left( E\right) \rightarrow M\times M\) be the associated vector bundle over \(M\times M\) with fibers, \({\text {Hom}}_{\left( q,p\right) }\left( E\right) :=L\left( E_{p},E_{q}\right) \) for \(\left( q,p\right) \in M\times M,\) where \(L\left( E_{p},E_{q}\right) \) denote the set of all linear transformations from \(E_{p}\) to \(E_{q}.\) A smooth local section U\(\in \Gamma \left( {\text {Hom}}\left( E\right) \right) \) with domain \(\mathsf {D}\) (i.e. \(U\left( q,p\right) \in L\left( E_{p} ,E_{q}\right) \) for all \(\left( q,p\right) \in \mathsf {D}\)) is called a parallelism if \(U\left( p,p\right) = {\text {id}}_{p}\). If U is only defined on a local diagonal domain, we refer to U as a local parallelism.
Example 87
(Parallel translation and parallelisms) One natural example of a parallelism when \(\left( M,g\right) \) is a Riemannian manifold and E is equipped with a covariant derivative, \(\nabla ^{E},\) is to define
where \(p,q\in M\) are “close enough” so there is a unique vector \(v_{p}\) with minimum length such that \(q=\exp _{p}\left( v_{p}\right) \) and \(//_{\left( \cdot \right) }^{E}\) denotes the parallel translation operator on E relative to \(\nabla ^{E}.\) For our purposes below E will be a bundle associated to TM and \(\nabla ^{E}\) will be the induced connection on this bundle associated to the Levi–Civita covariant derivative on \(\left( M,g\right) .\)
Example 88
(Charts and parallelisms) Each chart \((\Psi ,\mathcal {U})\) induces a local parallelism on \((T^{*}M)^{\otimes \ell }\) for any \(\ell \in \mathbb {N}\) as follows. If \(A\in (T_{p}^{*}M)^{\otimes \ell }\) is expressed as
then we define \(U^{\Psi }(q,p)A\in T_{q}^{*}M^{\otimes \ell }\) by
In other words, \(U^{\Psi }(q,p)\) is uniquely determined by requiring
for all \(q\in \mathcal {U}\) and \(1\le i_{1},i_{2},\dots i_{\ell }\le d.\) [This example is basically a special case of Example 87 where one takes \(\nabla \) to be the flat connection, \(D^{\Psi },\) defined in Definition 70.]
With the aid of a parallelism, we can now define the notion of \(\gamma \) – Hölder section, S, on E. In what follows we assume that E is equipped with a smoothly varying inner product, \(\left\langle \cdot ,\cdot \right\rangle _{E}.\) We do not necessarily assume that \(\nabla ^{E}\) is compatible with \(\left\langle \cdot ,\cdot \right\rangle _{E}\) or that \(U\left( p,q\right) \) is unitary for all \(\left( p,q\right) \in \mathsf {D}.\)
Lemma 89
Let S be a continuous section of a vector bundle E. Let \((U, \mathsf {D}), (U', \mathsf {D})\) be parallelisms on E. Then for every compactum \(K \subset \mathsf {D}\)
Proof
We work in a local trivialization. Let \(U, U': \mathbb {R}^{d}\times \mathbb {R}^{d}\rightarrow GL\left( \mathbb {R} ^{N}\right) \) be smooth functions such that \(U\left( x,x\right) , U'\left( x,x\right) ={\text {id}},\) which we view to be a parallelism on the trivial bundle, \(\mathbb {R}^{d} \times \mathbb {R}^{N}\) over \(\mathbb {R}^{d}.\) A continuous section of this bundle may be identified with a continuous function, \(S:\mathbb {R} ^{d}\rightarrow \mathbb {R}^{N}\) Then
The statement then follows from smoothness of \(U,U'\), the fact that they coincide at x, x and local boundedness of S. \(\square \)
Lemma 90
Let \(f\in C\left( M\right) ,\)\(\gamma >0\) and \(n=\lfloor \gamma \rfloor \in \mathbb {N}_{0}.\) Then \(f\in C^{\gamma }(M)\) (as in Definition 10) iff is f a n-times continuously differentiable function on M and for any (local) parallelism U on the vector bundle \(\Sigma ^{n}T^{*}M,\)\({\text {Sym}}[ \nabla ^n| f ]\) satisfies
Proof
Recall from Definition 10, that \(f\in C\left( M\right) \) is in \(C^\gamma (M)\) iff \(f\circ \Psi ^{-1}\in C^{\gamma }(\Psi (\mathcal {U}))\) for every coordinate chart \((\Psi ,\mathcal {U}).\) These conditions are equivalent to f being n-times continuously differentiable and the \(n^{\text {th}}\) – derivatives of \(f\circ \Psi ^{-1}\) being locally \(\left( \gamma -n\right) \)-Hölder on \(\Psi \left( \mathcal {U}\right) .\) The latter condition may be expressed as saying
where \(D=D^{\Psi }\) is the flat connection defined in Notation 70. From Lemma 72 and Corollary 73 we may express
where L is a linear differential operator of order at most \(n-1.\) As Lf is continuously differentiable it follows that
is continuously differentiable and vanishes at \(q=p\) and therefore (by the fundamental theorem of calculus)
From (41) and (42) it follows that (40) is equivalent to
Lastly using Lemma 89 we conclude that the estimates in (43) and (39) are also equivalent. \(\square \)
Theorem 91
Fix \(n\in \mathbb {N}_{0}\) and construct \(\mathcal {T}\) and \((\Pi ,\Gamma )\) as above. Then \(\mathcal {T}\) is a regularity structure (in the sense of Definition 13) and \((\Pi ,\Gamma )\) is a model of transport precision \(n+1\) (in the sense of Definition 14).
Proof
The fact that \(\mathcal {T}\) is a regularity structure is immediate. Let us now set \(\delta _M=\delta \) to be the injectivity radius of M and for \(q\in M,\) let \(U_{q}:=\exp _{q}( B_{T_q M}(0_q, \delta _M) ).\)
We have to check that
The homogeneity estimate, \(|\langle \Pi _{p}\tau ,\varphi _{p}^{\lambda } \rangle |\lesssim \lambda ^{\ell }\), for \(\tau \in \mathcal {T}_{\ell }|_{p}\) follows from the fact that \(\Pi _{p}\tau \) is a monomial of order \(\ell \) in \(\exp _{p}^{-1} \)-coordinates. Lemma 84 gives the transport precision, i.e.
Let D be the covariant derivative induced by the chart \(\exp _{q}^{-1}\). Using Lemma 72 we get
and hence \(||\Gamma _{p\leftarrow q}\tau _{\ell }||_{m}\lesssim d(p,q)^{\ell -m}\), which finishes the proof. \(\square \)
We are finally able to characterize \(C^{\gamma }(M)\) in terms of the “polynomial” regularity structure.
Theorem 92
Let \(\gamma \in (0,\infty ){\setminus }\mathbb {N}\) and \(f: M \rightarrow \mathbb {R}\) a continuous function. Then, \(f\in C^{\gamma }\left( M\right) \) if and only if there is \(\hat{f}\in \mathscr {D}^{\gamma }(M,\mathcal {T})\)Footnote 13 with \(\hat{f}_{0}\left( p\right) =f\left( p\right) \). In that case,
Proof
\(\left( \implies \right) \) Let \(f\in C^{\gamma }\left( M\right) \) and define
i.e. \(\hat{f}_{\ell }\left( p\right) :={\text {Sym}}[\nabla ^{\ell }|_{p}f]\) for \(0\le \ell \le \lfloor \gamma \rfloor =:n.\) We have to check that \(\hat{f}\in \mathscr {D}^{\gamma }(M, \mathcal {T})\), i.e. for all \(\ell \le \lfloor \gamma \rfloor \) and \(d(p,q) < \delta \)
or equivalently, using the definition of \(\Gamma _{q \leftarrow p}\), if
we must show
\(\ell =n\):
Recall \(\gamma -n\in (0,1]\). Now the term to bound in (44) reads as
By Lemma 80
and since the expression is smooth in q we can focus on
Define on the vector bundle \(\Sigma ^{n}T^{*}M\) the parallelism
Then by Lemma 90
so for \(\ell = n\) we are done.
\(\ell = 0, \dots , n - 1\): We need to show (44). It is enough to bound for \(w \in T_{p} M\), with \(v := \exp ^{-1}_{p}(q)\),
Here \(//_{t}( \gamma _{v} ) : T_{\gamma _{v}(0)} M \rightarrow T_{\gamma _{v}(t)} M\) denotes the parallel transport along \(\gamma _{v}(t) := \exp _{p}(tv)\).
For this purpose, define
and \(F:=f-g.\) Since \(W_{t}\) and \(\dot{\gamma }_{v}\left( t\right) \) are parallel along \(\gamma _{v}\left( t\right) \) it follows that
Therefore by Taylor’s theorem and the fact that \(\nabla ^{m}|_{p} F_{p}=0\) for \(0\le m\le n\),Footnote 14 we have
Since g is smooth we apply the fundamental theorem of calculus to find
Using this estimate, it follows that
Since
As shown in the step \(\ell = n\), we then get
and hence
Plugging this estimate back into (46) shows,
which completes the proof of (44).
\(\left( \Longleftarrow \right) \)
Recall that \(\gamma \in (n,n+1]\), for some \(n \in \mathbb {N}_{0}\).
Step 1: We will show that f is n-times differentiable and \(\frac{1}{\ell !} {\text {Sym}}[\nabla ^{\ell } f] = \hat{f}_{\ell }\) for \(\ell =0,\dots ,n\). This will be done by induction.
So assume for some \(\ell = 0,\dots ,n-1\) we know that
-
f is \(\ell \)-times differentiable
-
\(\frac{1}{i!} {\text {Sym}}[\nabla ^{i} f ] = \hat{f}_{i}, \quad i=0,\dots ,\ell \)
By Taylor’s theorem (Theorem 62)
Now by assumption
where
Hence
Plugging this into (47) and using the fact that \(|\dot{\gamma } _{v}(t)|=|v|\) we get
Now, since g is smooth and \(\frac{\nabla }{dt}\dot{\gamma }_{v}\left( t\right) =0,\) we have
and therefore by Taylor’s theorem (in one variable) together with Lemma 57
A simple integration by parts argument shows
Combining (48), (49) and (50), we get
As \(v\mapsto \exp _{p}\left( v\right) \) is a local diffeomorphism, it now follows from (51) that f is \(\ell +1\) times differentiable at p and moreover since,
we may conclude, using Lemma 57 that
Then by Remark 59 it follows that
Step 2: So far we have shown that f is n-times continuously differentiable and that \({\text {Sym}}[ \nabla ^\ell |_p f ] = \hat{f}_\ell (p)\) for \(\ell =0,\dots ,n\). Then with U defined in (45) we have
The second to last term is of order \(d(q,p)^{\gamma -n}\) by assumption. Moreover, for \(\ell \le n-1\), by Corollary 80, we have \(\nabla ^n|_p \Pi _p \hat{f}_\ell (p) = 0\). Hence the last term is of order \(d(q,p) \lesssim d(q,p)^{\gamma -n}\). By Lemma 90 we hence get that \(f \in C^\gamma (M)\). \(\square \)
9.3 A reformulation: the jet bundle
In this section we briefly outline that the “polynomial” regularity structure is in fact (isomorphic to) the jet bundle. In anticipation of possible future work with vector bundle valued equations, we formulate this in the general setting of a vector bundle \(E\overset{\pi }{\rightarrow }M\) with model fiber being a real finite dimensional vector space V. The connection to the previous sections is proven in Theorem 99. We then conclude by showing Taylor’s theorem in this setting, Theorem 103. We leave the complete construction of a polynomial regularity structure and its model in this fibered setting to future work. For background on vector bundles, we refer to [13, Chapter 10] and [16].
For each \(m\in M,\) let \(\Gamma \left( m\right) \) be the germ of \(C^{\infty }\)-local sections of E whose domain contains m. Fixing an integer \(n\in \mathbb {N}_{0},\) we define an equivalence relation on \(\Gamma \left( m\right) \) as follows. Let \(\left( x,u\right) \) be a chart and local frame such that \(m\in {\text {dom}}\left( x\right) ={\text {dom}}\left( u\right) \). We say \(S,T\in \Gamma \left( m\right) \) are equivalent and write \(S\overset{n}{\sim }T\) provided
where \(u^{-1}\left( p\right) :=u\left( p\right) ^{-1}:E_{p}\rightarrow V\) is the inverse of the linear operator, \(u\left( p\right) :V \rightarrow E_{p}.\) It is well known and easy to check that “\(\overset{n}{\sim } \)” is an equivalence relation which (by the chain and product rules) is independent of the choice of chart and local frame \(\left( x,u\right) \) as above.
The equivalence relation in (52) may also be written as \(S\overset{n}{\sim }T\) provided
where for an open subset, \(U\subset \mathbb {R}^{d},\)\(a\in U,\) and \(g\in C^{\infty }\left( U,V\right) ,\)\(\left( D^{k}g\right) \left( a\right) \) is the k-linear form on \(\mathbb {R}^{d}\) defined by
As mixed partial derivatives commute, \(\left( D^{k}g\right) \left( a\right) \) is symmetric and hence \(\left( D^{k}g\right) \left( a\right) \) is completely determined by its values on the diagonal, \(v_{1}=\dots =v_{k}=v\) and thus we may also write \(S\overset{n}{\sim }T\) provided
Definition 93
Given \(n\in \mathbb {N}_{0},\)\(m\in M\), and \(S\in \Gamma \left( m\right) ,\) let \(j_{m}^{n}S\) be the n-jet of S at m defined to be the equivalence class
The n-jet bundle of \(E\mathbf {,}\) denoted by \(J^{n}\left( \pi \right) ,\) is the vector bundle whose fiber over \(m\in M\) is \(J_{m} ^{n}\left( \pi \right) :=\left\{ j_{m}^{n}S:S\in \Gamma \left( m\right) \right\} .\)
We will identity \(J^{n}\left( \pi \right) \) with the vector bundle \(\oplus _{k=0}^{n}\Sigma ^{k}\left( T^{*}M\right) \otimes E.\) The identification that we will give is not canonical but will depend on choosing covariant derivatives, \(\nabla ^{E}\) and \(\nabla ^{M}\) on E and TM respectively. We now fix such a pair of covariant derivatives. We denote by \(\frac{\nabla ^{E}}{dt}\) (resp. \(\frac{\nabla ^{M}}{dt}\)) the covariant derivative of sections of E (resp. TM) along curves in M.
Definition 94
For \(v\in T_{m}M,\) let \(\sigma _{v}\left( t\right) :=\exp \left( tv\right) :=\exp ^{\nabla ^{M}}\left( tv\right) \) so that
By the inverse function theorem, there exists an open ball \(B\subset T_{m}M\) centered at \(0_{m}\in T_{m}m\) such that \(U:=\exp \left( B\right) \) is an open neighborhood of m, and \(\exp :B\rightarrow U\) is a diffeomorphism.
Definition 95
Continuing the notation above, let \(x:=\exp |_{U} ^{-1}:U\rightarrow B\subset T_{m}M\cong \mathbb {R}^{d}\) and for \(p=\exp \left( v\right) \in U\) with \(v=x\left( p\right) \) we let
Hence \(\left( x,U\right) \) is chart on M centered at m and u is local frame defined on \(U\subset M.\) [Note that \(\sigma _v\) and x both depend on \(\nabla ^M\) and u depends on both \(\nabla ^M\) and \(\nabla ^E\) even though this dependence is being suppressed from the notation.]
The next proposition is the key to coordinate free description of “\(\overset{n}{\sim }\)”.
Proposition 96
Let \(k\in \mathbb {N}_{0},\)\(m\in M,\)\(\nabla ^{E}\) and \(\nabla ^{M}\) be any covariant derivatives on E and TM respectively, and \(\left( x,u\right) \) be the chart/frame as in Definition 95. Then for any \(S\in \Gamma \left( m\right) ,\)
[Note that by construction \(x\left( m\right) =0_{m}\in T_{m}M.]\)
Proof
Let \(v\in T_{m}M\) be given and then choose \(\delta \in \left( 0,1\right) \) such that \(\sigma _{v}\left( t\right) \) exists for \(t\in \left[ 0,\delta \right] .\)Since \(\sigma _{tv}\left( s\right) =\sigma _{v}\left( ts\right) ,\)\(\sigma _{tv}|_{\left[ 0,\delta \right] }\) is a reparametrization of \(\sigma _{v}\ \)on \(\left[ 0,t\delta \right] ,\) we may conclude that
Using this observation along with \(x^{-1}\left( tv\right) =\exp \left( tv\right) =\sigma _{v}\left( t\right) ,\) we find
\(\square \)
The following is now an immediate corollary of Proposition 96.
Corollary 97
If \(n\in \mathbb {N}_{0},\)\(m\in M,\)\(\nabla ^{E}\) and \(\nabla ^{M}\) be any covariant derivatives on E and TM respectively, and \(S,T\in \Gamma \left( m\right) ,\) then \(S\overset{n}{\sim }T\) iff
Definition 98
If \(S\in \Gamma \left( m\right) \) and \(n\in \mathbb {N}_{0},\) let \(\mathrm {tay}_{m}^{n}\left( S\right) :T_{m}M\rightarrow E_{m}\) be the function defined for all \(v\in T_{m}M\) by
According to Proposition 96, we may rewrite (53) as
which shows that \(\mathrm {tay}_{m}^{n}\left( S\right) \in \mathcal {P} _{n}\left( T_{m}M,E_m\right) ,\) the linear space of degree n-polynomial on \(T_{m}M\) with values in \(E_{m}.\) Let us further note that \(D^{k}\left[ \left( u^{-1}S\right) \circ x^{-1}\right] \left( 0\right) \in \Sigma ^{k}\left[ T_{m}^{*}M\right] \) is uniquely determined by
where \(v\rightarrow \frac{1}{k!}\left( \partial _{v}^{k}\left[ \left( u^{-1}S\right) \circ x^{-1}\right] \right) \left( 0\right) \) is the k-homogeneous part of \(\mathrm {tay}_{m}^{n}\left( S\right) .\) Hence we may view \(\mathrm {tay}_{m}^{n}\left( S\right) \) as taking values in \(\Sigma ^{k}\left( T_{m}^{*}M\right) \otimes E_{m}.\) Finally observe that, by Corollary 97, if \(S\overset{n}{\sim }T,\) then \(\mathrm {tay}_{m} ^{n}\left( S\right) =\mathrm {tay}_{m}^{n}\left( T\right) ,\) i.e. we may view \(\mathrm {tay}_{m}^{n}\left( S\right) \) to be a function of \(j_{m}^{n}S\) rather than of S. This simply reflects the fact that if L is any n-order differential operator on \(\Gamma \left( E\right) \) then \(\left( LS\right) \left( m\right) =\left( LT\right) \left( m\right) \) whenever \(S\overset{n}{\sim }T.\)
Theorem 99
To each \(n\in \mathbb {N}_{0},\) the jet bundle \(J^{n}\left( \pi \right) \) is non-canonically isomorphic to \(\oplus _{k=0}^{n}\left[ \Sigma ^{k}\left( T^{*}M\right) \otimes E\right] .\) In more detail, if \(\nabla ^{E}\) and \(\nabla ^{M}\) are covariant derivatives on E and TM respectively and \(m\in M,\) then the associated Taylor map,
is a linear isomorphism of vector spaces.
Proof
In the lead up to the statement of the theorem we have already shown \(\mathrm {tay}_{m}^{n}\) in (54) is a well defined linear map and that its kernel has dimension zero. So to finish the proof we need only show \(\mathrm {tay}_{m}^{n}\) is surjective. To this end let \(Q\in \mathcal {P}_{n}\left( T_{m}M,E\right) \) which we decompose and let
be the k-homogeneous part of Q. By Taylor’s theorem, we know that \(Q\left( v\right) =\sum _{k=0}^{n}Q_{k}\left( v\right) .\) Using \(\left( x,u\right) \) from Definition 95, we then let
Then \(S\in \Gamma \left( m\right) \) and moreover for each \(0\le k\le n\) and \(v\in T_{m}M,\)
Thus it follows that
which shows \(\mathrm {tay}_{m}^{n}\) is surjective and completes the proof. \(\square \)
In order to write out \(\mathrm {tay}_{m}^{n}\) more explicitly, let \(\nabla \) denote the covariant derivative constructed on any of the bundles \(\left[ T^{*}M\right] ^{\otimes k}\otimes E\) which is constructed from \(\nabla ^{E}\) and \(\nabla ^{M}\) in such a way that the product rule holds. Note that \(\nabla ^k S\) is then a section of \([T^* M]^{\otimes k} \otimes E\), so for \(v \in T_m M\)
is an element of \(E_m\). This is consistent with earlier notation, where for \(E = M \times \mathbb {R}\) the pairing was real valued.
Proposition 100
If \(S\in \Gamma \left( m\right) \) and \(v\in T_{m}M,\) then
where \(\nabla ^{k}S=\overset{k\text {-times}}{\overbrace{\nabla \cdots \nabla }}S.\)
Proof
We will show by induction that
The case \(k=0\) is trivial, and the case \(k=1\) holds, since
For the induction step we compute;
wherein we have used,
as \(\sigma _{v}\left( \cdot \right) \) is a \(\nabla ^{M}\) -geodesic and the induction step in complete. Evaluating (55) at \(t=0\) then shows
as desired. \(\square \)
In the case of the trivial bundle \(E = M \times \mathbb {R}\), Theorem 99 shows that the jet bundle is just another representation of the “polynomial” regularity structure of the preceeding section. We finish this section, by showing Taylor’s theorem in the setting of general vector bundle presented here.
Definition 101
For \(n\in \mathbb {N}_{0},\)\(m\in M,\) and \(S\in \Gamma \left( m\right) ,\) define
for \(p\in U=\exp \left( B\right) \) as in Definition 95.
Remark 102
Taking \(E = M \times \mathbb {R}\), this is consistent with Definition 65.
Theorem 103
(Taylor’s Theorem) If \(S\in \Gamma \left( E\right) \) is a smooth section, \(m\in M,\) and \(n\in \mathbb {N}_{0},\) then
where
Alternatively we may express (56) as (recall Definition 95 for the definition of u)
Proof
For \(v\in B\subset T_{m}M,\) let \(s\left( t\right) :=//_{t}\left( \sigma _{v}\right) ^{-1}S\left( \sigma _{v}\left( t\right) \right) \in E_{m}.\) Applying Taylor’s theorem with integral remainder to s implies
and hence
where
Combining (57) and (58) with the identity,
completes the proof. \(\square \)
Notes
The support can be empty, in which case such an inclusion holds trivially.
The particular choice of \(\delta _\mathcal {K}/ 4\)-, \(\delta _\mathcal {K}/2\)-balls in the assumptions is needed in order to be able to invoke the norm here.
Compare also [10, Theorem 2.10] for a concise presentation of the (wavelet) techniques involved in its proof.
Recall from the beginning of this section that \(f_\alpha , f_{2+2\alpha }\) are real-valued and \(f_{1 + \alpha }\) is a section of \(T^* M\).
Recall that \(\mathcal {R}_t\) is the reconstruction operator of Theorem 23 associated to the model \((\Pi ^t,\Gamma ^t)\).
Assuming that one builds a regularity structure including space and time.
In coordinates,
$$\begin{aligned} \nabla ^2 {\mathsf {p}}^n_{t-s}(\gamma (r), \cdot ) = \left( \partial _{ij} {\mathsf {p}}^n_{t-s}(\gamma (r), \cdot ) - \sum _k\Gamma _{ij}^k \partial _k {\mathsf {p}}^n_{t-s}(\gamma (r), \cdot ) \right) dx^i \otimes dx^j, \end{aligned}$$where \(\Gamma \) are the Christoffel symbols. This gives the quadratic factor in \(|\dot{\gamma }(r)| = d(p,q)\). The blowup in \(t-s\) follows from an application of Lemma 44 (i), (ii) to the components here.
In the notation of [9, Proposition 3.32], \(\varphi ^n_z\) stands for \(2^{-nd/2}\varphi _z^{2^{-n}}\).
In particular, almost surely, for all \(\varphi \in C^{\infty }(M) \) and \(p\in M,\)\( t\mapsto \langle {\Pi ^t_p(\tau ),\varphi } \rangle \) is measurable and bounded.
Where we denote the distribution \(\langle \mathbb {E}[ I_{p,s} ], \varphi \rangle := \mathbb {E}[ \langle I_{p,s}, \varphi \rangle ]\).
Here we recall the notation of \(\hat{f}_\ell (x)\) as the component of \(\hat{f}(x)\) on the \(\ell \)-th homogeneity, i.e. the coefficient in front of \(\varvec{X}^{\ell }\).
In general, \(\nabla \) can be any affine connection.
The space of modelled distributions was defined in Definition 18.
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Acknowledgements
Open access funding provided by Max Planck Society. The authors wish to thank the referees for useful comments and suggestions, especially for pointing out the relation with jet bundle. The first author was supported in part by RTG 1845 and EPSRC grant EP/I03372X/1. The second author has received funding by the DAAD P.R.I.M.E. program. He would like to thank Giuseppe Cannizzaro and Konstantin Matetski for discussion on the Schauder estimates.
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Dahlqvist, A., Diehl, J. & Driver, B.K. The parabolic Anderson model on Riemann surfaces. Probab. Theory Relat. Fields 174, 369–444 (2019). https://doi.org/10.1007/s00440-018-0857-6
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DOI: https://doi.org/10.1007/s00440-018-0857-6