Discretisation of regularity structures

TL;DR

This paper develops a general framework to extend the theory of regularity structures to discretized stochastic PDEs, accommodating various discretisation methods without focusing on a specific procedure.

Contribution

It introduces a flexible approach that applies regularity structures to discretizations using a scale parameter and a black box for sub-scale behavior, broadening applicability.

Findings

Framework successfully extends regularity structures to discretized PDEs

Applicable to various discretisation schemes without specific assumptions

Facilitates analysis of stochastic PDEs in discrete settings

Abstract

We introduce a general framework allowing to apply the theory of regularity structures to discretisations of stochastic PDEs. The approach pursued in this article is that we do not focus on any one specific discretisation procedure. Instead, we assume that we are given a scale $\varepsilon > 0$ and a "black box" describing the behaviour of our discretised objects at scales below $\varepsilon $.

Equations

$$\mathcal{L}u=F(u,\nabla u,\xi ),$$

$$\mathcal{L}{u}_{\epsilon }=F(u_{\epsilon },\nabla u_{\epsilon },{\xi }_{\epsilon }),$$

$${\mathcal{R}}^{\epsilon }{\mathcal{K}}_{\gamma }^{\epsilon }=K^{\epsilon }{\mathcal{R}}^{\epsilon }.$$

$$d_{\mathfrak{s}}(y,z)={\sup }_{i\in \{1,\dots ,d\}}|y_i-z_i{|}^{1/{\mathfrak{s}}_i}.$$

$$|({\iota }_{\epsilon }f)({\phi }_z^{\epsilon })|\lesssim {\epsilon }^{\alpha }\Vert f{\Vert }_{\alpha ;[{\phi }_z^{\epsilon }];z;\epsilon }.$$

$$\bigl{(}\iota_{\varepsilon}f\bigr{)}(\varphi)=\varepsilon^{|\mathfrak{s}|}\sum_{z\in\Lambda_{\varepsilon}^{d}}f(z)\,\varphi(z)\;\hbox{,}\quad\|f\|_{\alpha;\mathfrak{K}_{\varepsilon};z;\varepsilon}=\varepsilon^{-\alpha}\mathop{\mathrm{sup}}_{y\in\mathfrak{K}_{\varepsilon}\cap\Lambda_{\varepsilon}^{d}}|f(y)|\;.{\hbox{\rm(6)}\kern 5.69046pt\kern-5.69046pt}q:discrXeps{=}e5{6}$$

Full text

1. 1 Introduction

2. 1.1 Discussion

3. 1.2 Strategy

4. 1.3 Structure of the article

5. 1.4 Notation

6. 2 A framework for discretisations

1 Introduction

The theory of regularity structures is a framework developed by the second author in [Regularity] which allows to renormalise stochastic PDEs of the form

[TABLE]

that are ill-posed in the classical sense. Here, ${\mathcal{L}}$ is typically a parabolic differential operator, $\xi$ is a very irregular random input and $F$ is some local non-linearity. The naïve approach to study well-posedness of (1) would be to consider a sequence of smooth approximations

[TABLE]

and simply declare the limit of this sequence to be the solution to (1). However, in many cases it turns out that the sequence $u_{\varepsilon}$ either does not possess a limit or the limit is trivial. A way to circumvent this is to renormalise, which can be interpreted as a kind of “recentering” of the equation, see [BHZalg].

This amounts to allowing some of the constants appearing in the definition of $F$ appearing in the right hand side of (2) to be $\varepsilon$-dependent and more specifically to diverge as $\varepsilon$ tends to zero. The theory of regularity structures is very successful when applied to approximations $u_{\varepsilon}$ that are functions defined on $\mathbb{R}^{d}$, it however presently does not in general apply to discrete approximations of stochastic PDEs or to equations where the operator ${\mathcal{L}}$ itself is perturbed by a higher-order term, in such a way that its scaling properties are different at small scales (think for instance of ${\mathcal{L}}$ given by $\partial_{t}-\Delta$ and ${\mathcal{L}}_{\varepsilon}=\partial_{t}-\Delta+\varepsilon^{2}\Delta^{2}$). In the present article, we develop a general framework that is able to deal with these cases. The guiding principle is a separation of scales, i.e., above a certain scale $\varepsilon$ we show that the theory of regularity structures still applies, whereas scales below $\varepsilon$ are treated as a black box. As a result, our framework is flexible and does not rely on any specific discretisation procedure. More precisely the results in the article show that, given a stochastic PDE such that the theory of regularity structures can be applied to renormalise it, our present framework can be used to treat a large class of natural discretisations for it.

1.1 Discussion

Let us first comment on our motivation for this work as well as its relation to some previous work. One of the guiding questions in the study of particle systems is the question of universality, i.e., what are the characterising features for a class of particle systems that converge to the same limit under appropriate rescaling and how can one prove this convergence? The bulk of the literature is concerned with scaling limits of a fixed system which necessarily implies that the resulting limit is itself a scale-invariant object. The present article however is motivated by the situation where one considers a family of systems indexed by some parameter and one simultaneously tunes this parameter as one rescales the system. In this way, one typically obtains scaling limits that are not scale invariant themselves. One of the insights of the theory presented in [Regularity] is that they are however locally described by linear combinations of objects that are scale-invariant, but with different scaling exponents.

So far, the state of the art for answering such questions (in the second case where the limiting object can, at least formally, be described by a singular stochastic PDE) is to heavily rely on special features of the model(s) under consideration. In the type of situation of interest to us, some standard techniques consist of

Claim 1

extending the discrete equation to an equation defined on $\mathbb{R}^{d}$, see for instance the articles of Gubinelli and Perkowski [KPZreloaded], Mourrat and Weber [MourratWeber], Shen and Weber [ShenWeber], Zhu and Zhu [ZhuZhuNS, ZhuZhuPhi];

linearising the problem via a Hopf-Cole transformation, see for example the articles of Bertini and Giacomin [MR1462228], Corwin and Shen [CorwinShen], Corwin, Shen and Tsai [CorwinShenTsai], Corwin and Tsai [CorwinTsai], Dembo and Tsai [DemboTsai], Labbé [Labbe] in the setting of the KPZ equation;

transforming the equation into a martingale problem, see for instance the series of articles by Gonçalvez and Jara [GoncalvesJaraKPZ], Gubinelli and Jara [GubinelliJara], Gubinelli and Perkowski [GubinelliPerkowskienergy], in which the concept of energy solution is developed to study the stochastic Burgers / KPZ equation.

When applied in the correct context these techniques can be very powerful. The drawback however is that they are often quite sensitive to small perturbations of the model. Another problem of analytical techniques like regularity structures or paraproducts is that while they provide a rather clean and general-purpose toolbox in the continuum, their extensions to the discrete setting require purpose-built modifications that are much less reusable. It is therefore desirable to have a robust theory at hand which is insensitive to the details of the underlying discrete setting, and it is the goal of the present article to make a first attempt at developing such a theory.

Another motivation to introduce the framework developed here stems from the fact that a common way to derive properties of a stochastic PDE is to approximate it by discrete systems for which the desired property holds and then show that it remains stable under passage to the limit. This methodology was for instance successfully applied by Hairer and Matetski [MatetskiDiscrete] to prove that the $\Phi_{3}^{4}$-measure built in [GlimmJaffe, Fel74] is invariant for the $\Phi_{3}^{4}$-equation. In the specific case where the solution to the stochastic PDE at hand is a function of time, [MatetskiDiscrete] developed a framework adapting the theory of regularity structures to allow for certain spatial discretisations.

1.2 Strategy

The main idea to accommodate a large class of discretisations of stochastic PDEs is to consider the behaviour below scale $\varepsilon$ as encoded in a “black box”. In order to describe this, our main ingredient is a sequence of linear spaces ${\mathcal{X}}_{\varepsilon}$ that can be viewed as subspaces of ${\mathcal{D}}^{\prime}(\mathbb{R}^{d})$, the space of distributions, and that possess a natural family of (extended) seminorms. We then work with stochastic PDEs of the type (2) such that $u_{\varepsilon}\in{\mathcal{X}}_{\varepsilon}$. Examples for ${\mathcal{X}}_{\varepsilon}$ include ${\mathcal{D}}^{\prime}(\mathbb{R}^{d})$ (thus recovering the original setting in [Regularity]), but also the space of functions defined on a discrete grid of mesh size $\varepsilon$, or simply some space of smooth functions. To cast this into the framework of regularity structures we first work with an abstract version of $u_{\varepsilon}$, i.e., we write $u_{\varepsilon}$ as a generalised Taylor expansion. Here, the monomials may represent the classical Taylor monomials or they may represent abstract expressions that are functions of $\xi$. The control of these monomials typically amount to the control of explicit stochastic objects and heavily depend on the choice of the equation at hand.

The “Taylor coefficients” can be thought of as the “derivatives” of an abstract version of a ${\mathcal{C}}^{\gamma}$-function and they are given by the solution of an abstract fixed point problem. We denote the class of abstract functions just described by ${\mathcal{D}}_{\varepsilon}^{\gamma}$. The subindex $\varepsilon$ indicates that the way regularity is measured above and below scale $\varepsilon$ may differ. Indeed, above scale $\varepsilon$ regularity is measured as in the continuous setting [Regularity], whereas below scale $\varepsilon$, besides some natural constraints, the way to measure regularity is not further determined. The idea of introducing ${\mathcal{D}}^{\gamma}$-spaces that depend on a parameter $\varepsilon$ is not new and already appeared in the works [KPZJeremy, Phi4Weijun]. In a sense the current article generalises some of the ideas developed there.

Note that the solution to (2) is a random space-time function / distribution, whereas the approach outlined above gives rise to an abstract “modelled distribution” $f\in{\mathcal{D}}_{\varepsilon}^{\gamma}$. To link the abstract object with a concrete object we construct a “reconstruction map” ${\mathcal{R}}^{\varepsilon}$. Unlike in [Regularity], we do not think of ${\mathcal{R}}^{\varepsilon}f$ as a distribution, but rather as an element of ${\mathcal{X}}_{\varepsilon}$, which could for example represent a space of functions defined on a discrete grid at scale $\varepsilon$. The way to construct ${\mathcal{R}}^{\varepsilon}$ is to postulate the existence of a map ${\mathcal{R}}^{\varepsilon}$ satisfying certain estimates on scales below $\varepsilon$ and to then show that analogous estimates automatically hold on all larger scales. Note that there may be several candidates for ${\mathcal{R}}^{\varepsilon}$ all satisfying the same quantitative estimates on small scales, so that in our context the reconstruction map is in general not uniquely defined. To proceed, we further need to define operations on ${\mathcal{D}}_{\varepsilon}^{\gamma}$ in order to actually construct abstract solutions to (2). In particular we define an abstract notion of convolution $\mathcal{K}_{\gamma}^{\varepsilon}$ against the Green function $K^{\varepsilon}$ of ${\mathcal{L}}$ in (2) (which is an operator mapping into ${\mathcal{X}}_{\varepsilon}$!) and we show that $\mathcal{K}_{\gamma}^{\varepsilon}$ satisfies a certain Schauder estimate. It turns out that unlike in the continuous setting [Regularity] it is not necessarily true that the convolution operator $\mathcal{K}_{\gamma}^{\varepsilon}$ can be defined in a natural way so that it intertwines with ${\mathcal{R}}^{\varepsilon}$ in the sense that

[TABLE]

We however argue that in many cases one can enforce (3) by tweaking the definition of $\mathcal{K}_{\gamma}^{\varepsilon}$. So far the strategy outlined above appears to only allow us to describe solutions to (2), but, as shown in [Regularity] and [BHZalg], the encoding of renormalisation procedures in the theory of regularity structures is of purely algebraic nature and does not depend on any specific discretisation procedure, see also Remark 1.1 below.

We finish with some concluding remarks that comment on how one does apply the theory developed in this article.

Remark 1.1

To apply the theory developed in this article to a concrete problem, an important ingredient is the construction of a suitable regularity structure and a suitable renormalised model. In the usual (continuous) case, a framework was built in [Regularity, Sec. 8], and further refined in [BHZalg, Ajay] that automatises this construction. To perform an analogous construction in the present context, one then needs an algebra structure on ${\mathcal{X}}_{\varepsilon}$, as well as a representation of the Taylor polynomials, which is often the case. The general analytical results of [Ajay] however fall out the scope of this article and would have to be adapted.

Remark 1.2

It was shown in [Regularity, Thm 10.7] (see also [NonGaussShen, Ajay]) that in order to obtain convergence of a sequence of models to a limiting model it is enough to obtain suitable bounds on sufficiently high moments for its terms of negative homogeneity. An important ingredient of the proof is the recursive definition of the regularity structure and the model, provided by the algebraic framework alluded to above. This is the same in the current case. Another ingredient is the identity $\Pi_{z}\tau={\mathcal{R}}f_{\tau,z}$, where $f_{\tau,z}(y)=\Gamma_{yz}\tau-\tau$ for any $\ell\in A$ and $\tau\in{\mathcal{T}}_{\ell}$ with positive homogeneity and $f_{\tau,z}(y)=\Gamma_{yz}\tau$ in case $\tau$ has negative homogeneity. Provided that the norm $|\!|\!|\cdot|\!|\!|_{\gamma;\mathfrak{K};\varepsilon}$ of Definition 2.1 below is chosen such that $|\!|\!|z\mapsto\Gamma_{zz^{\prime}}^{\varepsilon}\tau|\!|\!|_{\ell;\mathfrak{K};\varepsilon}=0$ for any $\ell\in A$, any $\tau\in{\mathcal{T}}_{\ell}$ and any $z^{\prime}\in\mathbb{R}^{d}$, the aforementioned identity follows from Assumption LABEL:a:rec. A further ingredient is the characterisation of the spaces of distributions under consideration by a wavelet basis. On scales larger than $\varepsilon$ a similar characterisation holds in the current case, see Definition LABEL:def:model below. On small scales, the situation depends on the choice of seminorms on ${\mathcal{X}}_{\varepsilon}$ introduced in Definition 2.1, but, in many cases of interest, we expect to have a choice of these seminorms which makes the corresponding bounds simple to verify. Finally, the last ingredient is the extension theorem [Regularity, Thm 5.14]. The analogous statement in the present setup is the content of Theorem LABEL:thm:extension. Thus, under the appropriate assumptions [Regularity, Thm 10.7] also holds in the framework constructed in this article.

1.3 Structure of the article

In Section 2 we develop the framework we will work with, in particular we introduce the sequence of spaces ${\mathcal{X}}_{\varepsilon}$, we define the notion of discrete models and the ${\mathcal{D}}_{\varepsilon}^{\gamma}$-spaces. In Sections LABEL:S3–LABEL:S6 we explain the main operations on these spaces.

1.4 Notation

Throughout this work $\mathfrak{s}=(\mathfrak{s}_{1},\ldots,\mathfrak{s}_{d})\in\mathbb{N}_{\geq 1}^{d}$ denotes a scaling of $\mathbb{R}^{d}$ and we associate to it the metric $d_{\mathfrak{s}}$ on $\mathbb{R}^{d}$ given by

[TABLE]

We sometimes use the notation $\|y-z\|_{\mathfrak{s}}$ in place of $d_{\mathfrak{s}}(y,z)$. Moreover, we let $|\mathfrak{s}|=\mathfrak{s}_{1}+\cdots+\mathfrak{s}_{d}$ and for a multiindex $k$ we use the notation $|k|_{\mathfrak{s}}=\sum_{i}\mathfrak{s}_{i}k_{i}$. Given a set $B\subset\mathbb{R}^{d}$ and $z\in\mathbb{R}^{d}$ we denote the distance of $z$ to $B$ with respect to the metric $d_{\mathfrak{s}}$ by $d_{\mathfrak{s}}(z,B)$. Given $\delta>0$ and $\varphi:\mathbb{R}^{d}\to\mathbb{R}$ we set ${\mathcal{S}}_{\mathfrak{s}}^{\delta}(z_{1},\ldots,z_{d})=(\delta^{-\mathfrak{s}_{1}}z_{1},\ldots,\delta^{-\mathfrak{s}_{d}}z_{d})$ and $({\mathcal{S}}_{\mathfrak{s},z}^{\delta}\varphi)(y)=\delta^{-|\mathfrak{s}|}\varphi({\mathcal{S}}_{\mathfrak{s}}^{\delta}(y-z))$. We also use occasionally the notation $\varphi_{z}^{\delta}$. Moreover, we also write $\varphi_{z}^{n}$ in place of $\varphi_{z}^{2^{-n}}$. For a compact subset $\mathfrak{K}$ of $\mathbb{R}^{d}$ (also written as $\mathfrak{K}\Subset\mathbb{R}^{d}$) we denote its $1$-fattening and $2$-fattening by $\bar{\mathfrak{K}}$ and $\bar{\bar{\mathfrak{K}}}$, respectively. We further denote the $d_{\mathfrak{s}}$-ball of radius $\delta$ around $z\in\mathbb{R}^{d}$ by ${\mathcal{B}}_{\mathfrak{s}}(z,\delta)$.

We occasionally use the notation $[\eta]$ to denote the support of the function $\eta$. In this article we use various notions of norms, seminorms and metrics on various spaces. To improve readability we inserted a norm index in the appendix listing all these distances.

Given a distribution $\xi$ and a test function $\varphi$, we use interchangeably the notations $\xi(\varphi)$, $\langle\xi,\varphi\rangle$, and $\int\varphi(x)\,\xi(dx)$ for the corresponding pairing.

Acknowledgements

MH gratefully acknowledges support by the Leverhulme Trust and by an ERC consolidator grant.

2 A framework for discretisations

Convention: From now on we assume that we are given a scaling $\mathfrak{s}$ of $\mathbb{R}^{d}$ and a regularity structure $\mathscr{T}=(A,{\mathcal{T}},G)$ containing the canonical $\mathfrak{s}$-scaled polynomials in $d$ indeterminates. We also fix a value $r>|\mathop{\mathrm{min}}A\,|$ and we denote by $\Phi$ the set of all functions $\varphi\in{\mathcal{C}}^{r}$ with $\|\varphi\|_{{\mathcal{C}}^{r}}\leq 1$ such that $\mathop{\mathrm{supp}}\nolimits\varphi\subset{\mathcal{B}}_{\mathfrak{s}}(0,1)$ and we simply say that “$\varphi$ is a test function” whenever $\varphi\in\Phi$. Given an element $\tau\in{\mathcal{T}}=\bigoplus_{\alpha\in A}{\mathcal{T}}_{\alpha}$, we write $\|\tau\|_{\alpha}$ for the norm of its component in ${\mathcal{T}}_{\alpha}$.

We build a general framework for allowing for discretisations of models and their convergence to continuous limiting models. Our construction relies crucially on the following notion.

Definition 2.1

A discretisation for the regularity structure $\mathscr{T}$ and scaling $\mathfrak{s}$ on $\mathbb{R}^{d}$ consists of the following data.

Claim 2

A collection of linear spaces ${\mathcal{X}}_{\varepsilon}$ with $\varepsilon\in(0,1]$ endowed with inclusion maps $\iota_{\varepsilon}\colon{\mathcal{X}}_{\varepsilon}\hookrightarrow{\mathcal{D}}^{\prime}(\mathbb{R}^{d})$, so that elements of ${\mathcal{X}}_{\varepsilon}$ can be interpreted as distributions.

Each ${\mathcal{X}}_{\varepsilon}$ admits a family of extended seminorms $\|\cdot\|_{\alpha;\mathfrak{K}_{\varepsilon};z;\varepsilon}$ (i.e., we do not exclude the possibility that $\|f\|_{\alpha;\mathfrak{K}_{\varepsilon};z;\varepsilon}=\infty$ for some $f\in{\mathcal{X}}_{\varepsilon}$) with $\alpha\in\mathbb{R}$, the $\mathfrak{K}_{\varepsilon}$ are compact subsets of $\mathbb{R}^{d}$ of diameter at most $2\varepsilon$ and $z\in\mathbb{R}^{d}$. We moreover assume that these seminorms are local in the sense that if $f,g\in{\mathcal{X}}_{\varepsilon}$ and $(\iota_{\varepsilon}f)(\varphi)=(\iota_{\varepsilon}g)(\varphi)$ for every $\varphi\in{\mathcal{C}}^{r}$ supported in $\mathfrak{K}_{\varepsilon}$, then $\|f-g\|_{\alpha;\mathfrak{K}_{\varepsilon};z;\varepsilon}=0$.

Uniformly over all $f\in{\mathcal{X}}_{\varepsilon}$, $z\in\mathbb{R}^{d}$, $\alpha\in\mathbb{R}$, and $\varphi\in\Phi$, one has the bound

[TABLE]

For any function $\Gamma\colon\mathbb{R}^{d}\times\mathbb{R}^{d}\to G$, there exists a family of extended seminorms $|\!|\!|\cdot|\!|\!|_{\gamma;\mathfrak{K};\varepsilon}$ on the space of functions $f:\mathbb{R}^{d}\to{\mathcal{T}}_{<\gamma}$ with $\gamma\in\mathbb{R}$, $\mathfrak{K}\Subset\mathbb{R}^{d}$ and $\varepsilon\in(0,1]$. For the same set of indices, and given two functions $\Gamma_{1},\Gamma_{2}:\mathbb{R}^{d}\times\mathbb{R}^{d}\to G$, there is a family of seminorms $|\!|\!|\cdot;\cdot|\!|\!|_{\gamma;\mathfrak{K};\varepsilon}$ on the space of pairs $(f,g)$, with $f,g:\mathbb{R}^{d}\to{\mathcal{T}}_{<\gamma}$. Both families of seminorms are assumed to depend only on the values of $f$ (and $g$ respectively) in a neighbourhood of size $\mathbf{c}\varepsilon$ around $\mathfrak{K}$, for $\mathbf{c}\geq 0$ a fixed constant.

Remark 2.2

The above way of introducing the seminorms in the fourth item may be a bit misleading. As we will see in the examples below, they indeed depend on the functions $\Gamma,\Gamma_{1}$, and $\Gamma_{2}$ introduced above, so that the correct way of denoting them would be $|\!|\!|\cdot|\!|\!|_{\gamma;\Gamma;\mathfrak{K};\varepsilon}$ and $|\!|\!|\cdot;\cdot|\!|\!|_{\gamma;\Gamma_{1},\Gamma_{2},\mathfrak{K};\varepsilon}$. However, for the sake of readability we omit these additional indices.

Remark 2.3

Assume that in the fourth item in the above definition one has $\Gamma_{1}=\Gamma_{2}$. In practice one then often has the relation $|\!|\!|f;g|\!|\!|_{\gamma;\mathfrak{K};\varepsilon}=|\!|\!|f-g|\!|\!|_{\gamma;\mathfrak{K};\varepsilon}$. See for instance the four examples introduced further below, where this is indeed the case.

To illustrate the setup described above, we mention the following four examples to which we will refer frequently in this work. We leave it as an exercise to verify that these are indeed examples of discretisations in the above sense. In all examples below $\mathfrak{K}_{\varepsilon}$ denotes a compact subset of $\mathbb{R}^{d}$ with diameter bounded by $2\varepsilon$.

The purely discrete case. Let $\Lambda_{\varepsilon}^{d}\subset\mathbb{R}^{d}$ be a locally finite set such that, for some constants $c,C\in(0,1]$, it holds that for every $z\in\Lambda_{\varepsilon}^{d}$, $\mathop{\mathrm{inf}}_{z\neq z^{\prime}\in\Lambda_{\varepsilon}^{d}}\|z-z^{\prime}\|_{\mathfrak{s}}\in[c\varepsilon,C\varepsilon]$. We then set ${\mathcal{X}}_{\varepsilon}=\mathbb{R}^{\Lambda_{\varepsilon}^{d}}$ with

[TABLE]