Quantitative stochastic homogenization and large-scale regularity

TL;DR

This work provides a comprehensive and simplified presentation of quantitative stochastic homogenization and large-scale regularity theory for elliptic equations, including new results and optimal estimates for various related problems.

Contribution

It introduces new simplified proofs and several novel results, such as optimal estimates for the Dirichlet problem and Green functions, advancing the understanding of stochastic homogenization.

Findings

Algebraic convergence rate for variational subadditive quantities

Large-scale Lipschitz and higher regularity estimates

Optimal quantitative estimates for homogenization error and two-scale expansion

Abstract

This is a preliminary version of a book which presents the quantitative homogenization and large-scale regularity theory for elliptic equations in divergence-form. The self-contained presentation gives new and simplified proofs of the core results proved in the last several years, including the algebraic convergence rate for the variational subadditive quantities, the large-scale Lipschitz and higher regularity estimates and Liouville-type results, optimal quantitative estimates on the first-order correctors and their scaling limit to a Gaussian free field. There are several chapters containing new results, such as: quantitative estimates for the Dirichlet problem, including optimal quantitative estimates of the homogenization error and the two-scale expansion; optimal estimates for the homogenization of the parabolic and elliptic Green functions; and $W^{1,p}$-type estimates for two-scale expansions.

Figures

Equations

$$-\nabla \cdot \left(\mathbf{a}(x)\nabla u\right)=f,$$

$$-\nabla \cdot \left(\mathbf{a}\left(\frac{x}{\epsilon }\right)\nabla u^{\epsilon }\right)=f,$$

$$-\nabla \cdot \left(\text{vbox}\text{hrule}height=0.7pt\mathbf{a}\nabla u\right)=f.$$

$${\partial }_{t}u-\nabla \cdot \mathbf{a}\nabla u=0.$$

$$\{\begin{array}{rlrl}& {\partial }_{t}u-\nabla \cdot \left(\mathbf{a}\nabla u\right)=0& \text{mbox}in& (0,\infty )\times {\mathbb{R}}^d,\\ & u(0,\cdot )=\nabla \cdot \mathbf{g}& \text{mbox}on& {\mathbb{R}}^d,\end{array}$$

$${\left|\xi \right|}^2⩽\xi \cdot \mathbf{a}(x)\xi ⩽\Lambda {\left|\xi \right|}^2,\forall \xi \in {\mathbb{R}}^d.$$

$$\Omega:=\left\{\mathbf{a}\,:\,\mathbf{a}\ \mbox{is a Lebesgue measurable map from ${\mathbb{R}^{d}}$ to $\mathbb{R}^{d\times d}_{\mathrm{sym}}$ satisfying\leavevmode\nobreak\ \eqref{e.ue}}\right\}.$$

$${\mathcal{F}}_U$$

$$\left\{\mathbf{a}\mapsto {\int }_{{\mathbb{R}}^d}{\mathbf{a}}_{ij}(x)\phi (x)dx:\phi \in C_c^{\infty }(U),i,j\in \{1,\dots ,d\}\right\}.$$

$$\left(T_y\mathbf{a}\right)(x):=\mathbf{a}(x+y),$$

$$\mathbb{P}\circ T_z=\mathbb{P}\text{mbox}foreveryz\in {\mathbb{Z}}^d.$$

$$\displaystyle\mathcal{F}_{U}\ \mbox{and}\ \mathcal{F}_{V}\ \ \mbox{are $\mathbb{P}$-independent for every pair $U,V\subseteq{\mathbb{R}^{d}}$}$$

$$\mathrm{dist}(U,V)⩾1$$

$$\mathbb{E}\left[X\right]:={\int }_{\Omega }X(\mathbf{a})d\mathbb{P}(\mathbf{a}).$$

$$\mathbb{P}[b(z)=0]=\mathbb{P}[b(z)=1]=\frac{1}{2},$$

$$\left\{\tilde{\mathbf{a}}\in {\mathbb{R}}_{\mathrm{sym}}^{d\times d}:\forall \xi \in {\mathbb{R}}^d,{\left|\xi \right|}^2⩽\xi \cdot \tilde{\mathbf{a}}\xi ⩽\Lambda {\left|\xi \right|}^2\right\}.$$

$$\mathbf{a}(x)={\mathbf{a}}_{b(z)}.$$

$$\mathbf{a}(x):=\mathbf{a}(z).$$

$$\mathbf{a}(x):=\{\begin{array}{rl}{\mathbf{a}}_0& \text{if }\mathrm{dist}(x,\Pi )⩽\frac{1}{2},\\ {\mathbf{a}}_1& \text{otherwise}.\end{array}$$

$$\mathbf{a}(x):=\{\begin{array}{rl}{\mathbf{a}}_0& \text{if there exists }(z,r)\in \Pi \text{ such that }|x-z|⩽r,\\ {\mathbf{a}}_1& \text{otherwise}.\end{array}$$

$$\mathbf{a}(x)=F((\mathsf{W}\ast \varphi )(x)).$$

$$A\in \mathcal{F}$$

$$\mathbb{P}\left[\underset{n\to \infty }{\mathrm{limsup}}\left|\mathbf{a}({\text{scaleobj}1.2\square }_n)-\text{vbox}\text{hrule}height=0.7pt\mathbf{a}\right|=0\right]=1.$$

$${\mathbf{P}}_x^{\mathbf{a}}\left[X_t\in A\right]={\int }_{A}P(t,x,y)dy,$$

$$\text{for every }x\in {\mathbb{R}}^d,t^{\frac{d}{2}}P(t,0,t^{\frac{1}{2}}x)\underset{t\to \infty }{\overset{\text{a.s.}}{\to }}\overline{P}(1,0,x),$$

$$U_r:=\left\{x\in U:\mathrm{dist}(x,\partial U)>r\right\}\text{mbox}and{U}^r:=\left\{x\in {\mathbb{R}}^m:\mathrm{dist}(x,U)<r\right\}.$$

$${\text{scaleobj}1.2\square }_m:={\left(-\frac{1}{2}{3}^m,\frac{1}{2}{3}^m\right)}^d\subseteq {\mathbb{R}}^d.$$

$${\nabla }^{k}f:={\left({\partial }_{i_1}\cdots {\partial }_{i_k}f\right)}_{i_1,\dots ,i_k\in \{1,\dots ,d\}}$$

$${\Vert u\Vert }_{C^{k,\alpha }(U)}:=\sum _{n=0}^k\underset{x\in U}{\sup }\left|{\nabla }^{n}u(x)\right|+{\left[{\nabla }^{k}u\right]}_{C^{0,\alpha }(U)}$$

$${\left[u\right]}_{C^{0,\alpha }(U)}:=\underset{x,y\in U,x\ne y}{\sup }\frac{\left|u(x)-u(y)\right|}{|x-y{|}^{\alpha }}.$$

$${\int }_{U}f:={\int }_{U}f(x)dx.$$

Full text

1. 1 Preface
2. 2 Assumptions and examples
3. 3 Frequently asked questions
4. 4 Notation

Chapter 1 Preface

Many microscopic models lead to partial differential equations with rapidly oscillating coefficients. A particular example, which is the main focus of this book, is the scalar, uniformly elliptic equation

[TABLE]

where the interest is in the behavior of the solutions on length scales much larger than the unit scale (the microscopic scale on which the coefficients are varying). The coefficients are assumed to be valued in the positive definite matrices, and may be periodic, almost periodic, or stationary random fields. Such equations arise in a variety of contexts such as heat conduction and electromagnetism in heterogeneous materials, or through their connection with stochastic processes.

To emphasize the highly heterogeneous nature of the problem, it is customary to introduce a parameter $0<\varepsilon\ll 1$ to represent the ratio of the microscopic and macroscopic scales. The equation is then rescaled as

[TABLE]

with the problem reformulated as that of determining the asymptotic behavior of $u^{\varepsilon}$, subject to appropriate boundary conditions, as $\varepsilon\to 0$.

It has been known since the early 1980s that, under very general assumptions, the solution $u^{\varepsilon}$ of the heterogeneous equation converges in $L^{2}$ to the solution $u$ of a constant-coefficient equation

[TABLE]

We call this the homogenized equation and the coefficients the homogenized or effective coefficients. The matrix

$\mathbf{a}$

  will depend on the coefficients $\mathbf{a}\left(\cdot\right)$ in a very complicated fashion: there is no simple formula for 

$\mathbf{a}$

  except in dimension $d=1$ and some special situations in $d=2$. However, if one is willing to perform the computational work of approximating the homogenized coefficients and to tolerate the error in replacing $u^{\varepsilon}$ by $u$, then there is a potentially huge payoff to be gained in terms of a reduction of the complexity of the problem. Indeed, up to a change of variables, the homogenized equation is simply the Poisson equation, which can be numerically computed in linear time and memory and is obviously independent of $\varepsilon>0$. In contrast, the cost of computing the solution to the heterogeneous equation explodes as $\varepsilon$ becomes small, and can be considered out of reach.

There is a vast and rich mathematical literature on homogenization developed in the last forty years and already many good expositions on the topic (see for instance the books [Allaire, BLP, BraidesDe, CD, DalMaso, JKO, KLO, PavStuart, Tartar]). Most of these works are focused on qualitative results, such as proving the existence of a homogenized equation which characterizes the limit as $\varepsilon\to 0$ of solutions. The need to develop efficient methods for determining

$\mathbf{a}$

  and for estimating the error in the homogenization approximation (e.g., $\|u^{\varepsilon}-u\|_{L^{2}}$) motivates the development of a quantitative theory of homogenization. However, until recently, nearly all of the quantitative results were confined to the rather restrictive case of periodic coefficients. The main reason for this is that quantitative homogenization estimates in the periodic case are vastly simpler to prove than under essentially any other hypothesis (even the almost periodic case). Indeed, the problem can be essentially reduced to one on the torus and compactness arguments then yield optimal estimates. In other words, in the periodic setting, the typical arguments of qualitative homogenization theory can be made quantitative in a relatively straightforward way.

This book is concerned with the quantitative theory of homogenization for nonperiodic coefficient fields, focusing on the case in which $\mathbf{a}(x)$ is a stationary random field satisfying quantitative ergodicity assumptions. This is a topic which has undergone a rapid development since its birth at the beginning of this decade, with new results and more precise estimates coming at an ever accelerating pace. Very recently, there has been a convergence toward a common philosophy and set of core ideas, which have resulted in a complete and optimal theory. The purpose of this book is to give this theory a complete and self-contained presentation.

We have written it with several purposes and audiences in mind. Experts on the topic will find new results as well as arguments which have been greatly simplified compared to the previous state of the literature. Researchers interested in stochastic homogenization will hopefully find a useful reference to the main results in the field and a roadmap to the literature. Our approach to certain topics, such as the construction of the Gaussian free field or the relation between Sobolev norms and the heat kernel, could be of independent interest to certain segments of the probability and analysis communities. We have written the book with newcomers to homogenization in mind and, most of all, graduate students and young researchers. In particular, we expect that readers with a basic knowledge of probability and analysis, but perhaps without expertise in elliptic regularity, the Gaussian free field, negative and fractional Sobolev spaces, etc, should not have difficulty following the flow of the book. These topics are introduced as they arise and are developed in a mostly self-contained way.

Before we give a summary of the topics we cover and the approach we take, let us briefly recall the historical and mathematical context. In the case of stationary random coefficients, there were very beautiful, soft arguments given independently in the early 1980s by Kozlov [K1], Papanicolaou and Varadhan [PV1] and Yurinskiĭ [Y0] which give proofs of qualitative homogenization under very general hypotheses. A few years later, Dal Maso and Modica [DM1, DM2] extended these results to nonlinear equations using variational arguments inspired by $\Gamma$-convergence. Each of the proofs in these papers relies in some way on an application of the ergodic theorem applied to the gradient (or energy density) of certain solutions of the heterogeneous equation. In order to obtain a convergence rate for the limit given by the ergodic theorem, it is necessary to verify quantitative ergodic conditions on the underlying random sequence or field. It is therefore necessary and natural to impose such a condition on the coefficient field $\mathbf{a}(x)$. However, even under the strongest of mixing assumptions (such as the finite range of dependence assumption we work with for most of this book), one faces the difficulty of transferring the quantitative ergodic information contained in these strong mixing properties from the coefficients to the solutions, since the ergodic theorem is applied to the latter. This is difficult because, of course, the solutions depend on the coefficient field in a very complicated, nonlinear and nonlocal way.

Gloria and Otto [GO1, GO2] were the first to address this difficulty in a satisfactory way in the case of coefficient fields that can be represented as functions of countably many independent random variables. They used an idea from statistical mechanics, previously introduced in the context of homogenization by Naddaf and Spencer [NS], of viewing the solutions as functions of these independent random variables and applying certain general concentration inequalities such as the Efron-Stein or logarithmic Sobolev inequalities. If one can quantify the dependence of the solutions on a resampling of each independent random variable, then these inequalities immediately give bounds on the fluctuations of solutions. Gloria and Otto used this method to derive estimates on the first-order correctors which are optimal in terms of the ratio of length scales (although not optimal in terms of stochastic integrability).

The point of view developed in this book is different and originates in works of Armstrong and Smart [AS], Armstrong and Mourrat [AM], and the authors [AKM1, AKM2]. Rather than study solutions of the equation directly, the main idea is to focus on certain energy quantities, which allow us to implement a progressive coarsening of the coefficient field and capture the behavior of solutions on large—but finite—length scales. The approach can thus be compared with renormalization group arguments in theoretical physics. The core of the argument is to establish that on large scales, these energy quantities are in fact essentially local, additive functions of the coefficient field. It is then straightforward to optimally transfer the mixing properties of the coefficients to the energy quantities and then to the solutions.

The quantitative analysis of the energy quantities is the focus of the first part of the book. After a first introductory chapter, the strategy naturally breaks into several distinct steps:

• •

Obtaining an algebraic rate of convergence for the homogenization limits, using the subadditive and convex analytic structure endowed by the variational formulation of the equation (Chapter LABEL:c.two). Here the emphasis is on obtaining estimates with optimal stochastic integrability, while the exponent representing the scaling of the error is suboptimal.

• •

Establishing a large-scale regularity theory: it turns out that solutions of an equation with stationary random coefficients are much more regular than one can show from the usual elliptic regularity for equations with measurable coefficients (Chapter LABEL:c.regularity). We prove this by showing that the extra regularity is inherited from the homogenized equation by approximation, using a Campanato-type iteration and the quantitative homogenization results obtained in the previous chapter.

• •

Implementing a modification of the renormalization scheme of Chapter LABEL:c.two, with the major additional ingredient of the large-scale regularity theory, to improve the convergence of the energy quantities to the optimal rate predicted by the scaling of the central limit theorem. Consequently, deriving optimal quantitative estimates for the first-order correctors (Chapter LABEL:c.A1).

• •

Characterizing the fluctuations of the energy quantities by proving convergence to white noise and consequently obtaining the scaling limit of the first-order correctors to a modified Gaussian free field (Chapter LABEL:c.gff).

• •

Combining the optimal estimates on the first-order correctors with classical arguments from homogenization theory to obtain optimal estimates on the homogenization error, and the two-scale expansion, for Dirichlet and Neumann boundary value problems (Chapter LABEL:c.twoscale).

These six chapters represent, in our view, the essential part of the theory. The first four chapters should be read consecutively (Sections LABEL:s.boundaryreg and LABEL:s.regularity.optimal can be skipped), while the Chapters LABEL:c.gff and LABEL:c.twoscale are independent of each other.

Chapter LABEL:c.CZ complements the regularity theory of Chapter LABEL:c.regularity by developing local and global gradient $L^{p}$ estimates ($2<p<\infty$) of Calderón-Zygmund-type for equations with right-hand side. Using these estimates, in Section LABEL:s.twoscale.p we extend the results of Chapter LABEL:c.twoscale by proving optimal quantitative bounds on the error of the two-scale expansion in $W^{1,p}$-type norms. Except for the last section, which requires the optimal bounds on the first-order correctors proved in Chapter LABEL:c.A1, this chapter can be read after Chapter LABEL:c.regularity.

Chapter LABEL:c.parabolic extends the analysis to the time-dependent parabolic equation

[TABLE]

The main focus is on obtaining a suboptimal error estimate for the Cauchy-Dirichlet problem and a parabolic version of the large-scale regularity theory. Here the coefficients $\mathbf{a}(x)$ depend only on space, and the arguments in the chapter rely on the estimates on first-order correctors obtained in Chapters LABEL:c.two and LABEL:c.regularity in addition to some relatively routine deterministic arguments. In Chapter LABEL:c.parabolic we also prove decay estimates on the elliptic and parabolic Green functions as well as on their derivatives, homogenization error and two-scale expansions.

In Chapter LABEL:c.semigroup, we study the decay, as $t\to\infty$, of the solution $u(t,x)$ of the parabolic initial-value problem

[TABLE]

where $\mathbf{g}$ is a bounded, stationary random field with a unit range of dependence. We show that the solution $u$ decays to zero at the same rate as one has in the case $\mathbf{a}=\mathsf{Id}$. As an application, we upgrade the quantitative homogenization estimates for the parabolic and elliptic Green functions to the optimal scaling (see Theorem LABEL:t.PtoPbar and Corollary LABEL:c.GtoGbar).

In Chapter LABEL:c.subadd-fitz, we show how the variational methods in this book can be adapted to non-self adjoint operators, in other words, linear equations with nonsymmetric coefficients. In Chapter LABEL:c.nonlinear we give a generalization to the case of nonlinear equations. In particular, in both of these chapters we give a full generalization of the results of Chapters LABEL:c.one and LABEL:c.two to these settings as well as the large-scale $C^{0,1}$ estimate of Chapter LABEL:c.regularity.

This version of the manuscript is essentially complete and, except for small changes and corrections and a modest expansion of Chapter LABEL:c.semigroup, we expect to publish it in close to its present form.

We would like to thank several of our colleagues and students for their helpful comments, suggestions, and corrections: Alexandre Bordas, Sanchit Chaturvedi, Paul Dario, Sam Ferguson, Chenlin Gu, Jan Kristensen, Jules Pertinand, Christophe Prange, Armin Schikorra, Charlie Smart, Tom Spencer, Stephan Wojtowytsch, Wei Wu and Ofer Zeitouni. We particularly thank Antti Hannukainen for his help with the numerical computations that generated Figure LABEL:f.correctors. SA was partially supported by NSF Grant DMS-1700329. TK was partially supported by the Academy of Finland and he thanks Giuseppe Mingione for the invitation to give a graduate course at the University of Parma. JCM was partially supported by the ANR grant LSD (ANR-15-CE40-0020-03).

There is no doubt that small mistakes and typos remain in the manuscript, and so we encourage readers to send any they may find, as well as any comments, suggestions and criticisms, by email. Until the manuscript is complete, we will keep the latest version on our webpages. After it is published as a book, we will also maintain a list of typos and misprints found after publication.

Scott Armstrong, New York

Tuomo Kuusi, Helsinki

Jean-Christophe Mourrat, Paris

Chapter 2 Assumptions and examples

We state here the assumptions which are in force throughout most of the book, and present some concrete examples of coefficient fields satisfying them.

Assumptions

Except where specifically indicated otherwise, the following standing assumptions are in force throughout the book.

We fix a constant $\Lambda>1$ called the ellipticity constant, and a dimension $d\geqslant 2$.

We let $\Omega$ denote the set of all measurable maps $\mathbf{a}(\cdot)$ from ${\mathbb{R}^{d}}$ into the set of symmetric $d\times d$ matrices, denoted by $\mathbb{R}^{d\times d}_{\mathrm{sym}}$, which satisfy the uniform ellipticity and boundedness condition

[TABLE]

That is,

[TABLE]

The entries of an element $\mathbf{a}\in\Omega$ are written as $\mathbf{a}_{ij}$, $i,j\in\{1,\ldots,d\}$.

We endow $\Omega$ with a family of $\sigma$-algebras $\left\{\mathcal{F}_{U}\right\}$ indexed by the family of Borel subsets $U\subseteq{\mathbb{R}^{d}}$, defined by

[TABLE]

The largest of these $\sigma$-algebras is denoted $\mathcal{F}:=\mathcal{F}_{{\mathbb{R}^{d}}}$. For each $y\in{\mathbb{R}^{d}}$, we let $T_{y}:\Omega\to\Omega$ be the action of translation by $y$,

[TABLE]

and extend this to elements of $\mathcal{F}$ by defining $T_{y}E:=\left\{T_{y}\mathbf{a}\,:\,\mathbf{a}\in E\right\}$.

Except where indicated otherwise, we assume throughout the book that $\mathbb{P}$ is a probability measure on the measurable space $(\Omega,\mathcal{F})$ satisfying the following two important assumptions:

• •

Stationarity with respect to $\mathbb{Z}^{d}$-translations:

[TABLE]

• •

Unit range of dependence:

[TABLE]

We denote the expectation with respect to $\mathbb{P}$ by $\mathbb{E}$. That is, if $X:\Omega\to\mathbb{R}$ is an $\mathcal{F}$-measurable random variable, we write

[TABLE]

While all random objects we study in this text are functions of $\mathbf{a}\in\Omega$, we do not typically display this dependence explicitly in our notation. We rather use the symbol $\mathbf{a}$ or $\mathbf{a}(x)$ to denote the canonical coefficient field with law $\mathbb{P}$.

Examples satisfying the assumptions

The simplest way to construct explicit examples satisfying the assumptions of uniform ellipticity (2.1), stationarity (2.5) and (2.6) is by means of a “random checkerboard” structure: we pave the space by unit-sized cubes and color each cube either white or black independently at random. Each color is then associated with a particular value of the diffusivity matrix. More precisely, let $(b(z))_{z\in\mathbb{Z}^{d}}$ be independent random variables such that for every $z\in\mathbb{Z}^{d}$,

[TABLE]

and fix two matrices $\mathbf{a}_{0},\mathbf{a}_{1}$ belonging to the set

[TABLE]

We can then define a random field $x\mapsto\mathbf{a}(x)$ satisfying (2.1) and with a law satisfying (2.5) and (2.6) by setting, for every $z\in\mathbb{Z}^{d}$ and $x\in z+\left[-\frac{1}{2},\frac{1}{2}\right)^{d}$,

[TABLE]

This example is illustrated on Figure 1. It can be generalized as follows: we give ourselves a family $(\mathbf{a}(z))_{z\in\mathbb{Z}^{d}}$ of independent and identically distributed (i.i.d.) random variables taking values in the set (2.8), and then extend the field $z\mapsto\mathbf{a}(z)$ by setting, for every $z\in\mathbb{Z}^{d}$ and $x\in z+\left[-\frac{1}{2},\frac{1}{2}\right)^{d}$,

[TABLE]

Another class of examples can be constructed using homogeneous Poisson point processes. We recall that a Poisson point process on a measurable space $(E,\mathcal{E})$ with (non-atomic, $\sigma$-finite) intensity measure $\mu$ is a random subset $\Pi$ of $E$ such that the following properties hold (see also [kingman]):

• •

For every measurable set $A\in\mathcal{E}$, the number of points in $\Pi\cap A$, which we denote by $N(A)$, follows a Poisson law of mean $\mu(A)$;

• •

For every pairwise disjoint measurable sets $A_{1},\ldots,A_{k}\in\mathcal{E}$, the random variables $N(A_{1})$, …, $N(A_{k})$ are independent.

Let $\Pi$ be a Poisson point process on ${\mathbb{R}^{d}}$ with intensity measure given by a multiple of the Lebesgue measure. Fixing two matrices $\mathbf{a}_{0},\mathbf{a}_{1}$ belonging to the set (2.8), we may define a random field $x\mapsto\mathbf{a}(x)$ by setting, for every $x\in{\mathbb{R}^{d}}$,

[TABLE]

This example is illustrated on Figure 2. More complicated examples can be constructed using richer point processes. For instance, in the construction above, each point of $\Pi$ imposes the value of $\mathbf{a}(x)$ in a centered ball of radius $1/2$; we may wish to construct examples where this radius itself is random. In order to do so, let $\lambda>0$, let $\mu$ denote a probability measure on $\left[0,\frac{1}{2}\right]$ (the law of the random radius), and let $\Pi$ be a Poisson point process on ${\mathbb{R}^{d}}\times\mathbb{R}$ with intensity measure $\lambda dx\otimes\mu$ (where $dx$ denotes the Lebesgue measure on ${\mathbb{R}^{d}}$). We then set, for every $x\in{\mathbb{R}^{d}}$,

[TABLE]

Minor variants of this example allow for instance to replace balls by random shapes, to allow the conductivity matrix to take more than two values, etc. See Figure 3 for an example.

Yet another class of examples can be obtained by defining the coefficient field $x\mapsto\mathbf{a}(x)$ as a local function of a white noise field. We refer to Definition LABEL:d.scalar.noise and Proposition LABEL:p.white.noise.exists for the definition and construction of white noise. For instance, given a scalar white noise $\mathsf{W}$, we may fix a smooth function $\phi\in C^{\infty}_{c}({\mathbb{R}^{d}})$ with support in $B_{1/2}$, a smooth function $F$ from ${\mathbb{R}^{d}}$ into the set (2.8), and define

[TABLE]

See Figure 4 for a representation of the scalar field $x\mapsto(\mathsf{W}\ast\phi)(x)$.

Chapter 3 Frequently asked questions

Where is the independence assumption used?

The unit range of dependence assumption (2.6) is obviously very important, and to avoid diluting its power we use it sparingly. We list here all the places in the book where it is invoked:

• •

The proof of Proposition LABEL:p.nuconv (which is made redundant by the following one).

• •

The proof of Lemma LABEL:l.spatavg (and the generalizations of this lemma appearing in Chapters LABEL:c.subadd-fitz and LABEL:c.nonlinear). This lemma lies at the heart of the iteration argument in Chapter LABEL:c.two, as it is here that we obtain our first estimate on the correspondence between spatial averages of gradients and fluxes of solutions. Notice that the proof does not use the full strength of the independence assumption, it actually requires only a very weak assumption of correlation decay.

• •

The last step of the proof of Theorem LABEL:t.subadd (and the generalizations of this theorem appearing in Chapters LABEL:c.subadd-fitz and LABEL:c.nonlinear). Here independence is used very strongly to obtain homogenization estimates with optimal stochastic integrability.

• •

The proof of Proposition LABEL:p.improvefluc in Section LABEL:s.fluctuations, where we control the fluctuations of the quantity $J_{1}$ inside the bootstrap argument.

• •

The proof of Proposition LABEL:p.d2vengeance in Section LABEL:s.twopoint, where we prove sharper bounds on the first-order correctors in dimension $d=2$.

• •

In Section LABEL:s.CLT, where we prove the central limit theorem for the quantity $J_{1}$. This can be considered a refinement of Proposition LABEL:p.improvefluc.

• •

In Section LABEL:s.sgp.opt in the proofs of Lemmas LABEL:l.SG.opt1 and LABEL:l.SG.opt4.

In particular, all of the results of Chapters LABEL:c.two and LABEL:c.regularity are obtained with only two very straightforward applications of independence.

Can the independence assumption be relaxed?

Yes. One of the advantages of the approach presented in this book is that the independence assumption is applied only to sums of local random variables. Any reasonable decorrelation condition or mixing-type assumption will give estimates regarding the stochastic cancellations of sums of local random variables (indeed, this is essentially a tautology). Therefore, while the statements of the theorems may need to be modified for weaker assumptions (for instance, the strong stochastic integrability results we obtain under a finite range of dependence assumption may have to be weakened), the proofs will only require straightforward adaptations. In fact, since we have only used independence in a handful of places in the text, enumerated above, it is not a daunting task to perform these adaptations. This is in contrast to alternative approaches in quantitative stochastic homogenization which use nonlinear concentration inequalities and therefore are much less robust to changes in the hypotheses.

The reason for formalizing the results under the strongest possible mixing assumption (finite range of dependence) rather than attempting write a very general result is, therefore, not due to a limitation of the arguments. It is simply because we favor clarity of exposition over generality.

Can the uniform ellipticity assumption be relaxed?

One of the principles of this book is that one should avoid using small-scale or pointwise properties of the solutions or of the equation and focus rather on large-scale, averaged information. In particular, especially in the first part of the book, we concentrate on the energy quantities $\nu$, $\nu^{*}$ and $J_{1}$ which can be thought of as “coarsened coefficients” in analogy to a renormalization scheme (see Remark LABEL:r.coarsened). The arguments we use adhere to this philosophy rather strictly. In particular, they are adaptable to situations in which the matrix $\mathbf{a}(x)$ is not necessarily uniformly positive definite, provided we have some quantitative information regarding the law of its condition number. This is because such assumptions can be translated into quantitative information about $J_{1}$ which suffices to run the renormalization arguments of Chapter LABEL:c.two. A demonstration of the robustness of these methods can also be found in [AD], which adapted Chapters LABEL:c.two and LABEL:c.regularity of this manuscript to obtain the first quantitative homogenization results on supercritical percolation clusters (a particularly extreme example of a degenerate environment).

Do the results in this book apply to elliptic systems?

Since the notation for elliptic systems is a bit distracting, we have decided to use scalar notation. However, throughout most of the book, we use exclusively arguments which also work for systems of equations (satisfying the uniform ellipticity assumption). The only exceptions are the last two sections of Chapter LABEL:c.parabolic and Chapter LABEL:c.semigroup, where we do use some scalar estimates (the De Giorgi-Nash $L^{\infty}$ bound and variations) which make it easier to work with Green functions. In particular, we claim that all of the statements and proofs appearing in this book, with the exception of those appearing in those two chapters, can be adapted to the case of elliptic systems with easy and straightforward modifications to the notation.

This book is written for equations in the continuum. Do the arguments apply to finite difference equations on $\mathbb{Z}^{d}$?

The techniques developed in the book are robust to the underlying structure of the environment on the microscopic scale. What is the important is that the “geometry” of the macroscopic medium is like that of ${\mathbb{R}^{d}}$ in the sense that certain functional inequalities (such as the Sobolev inequality) hold on large scales. In the case that ${\mathbb{R}^{d}}$ is replaced by $\mathbb{Z}^{d}$, the modifications are relatively straightforward: besides changes to the notation, there is just the slight detail that the boundary of a large cube has a nonzero volume, which creates an additional error term in Chapter LABEL:c.two causing no harm. If one has a more complicated microstructure like a random graph, such as a supercritical Bernoulli percolation cluster, it is necessary to first establish the “geometric regularity” of the graph in the sense that Sobolev-type inequalities hold on large scales. The techniques described in this book can then be readily applied: see [AD].

Can I find a simple proof of qualitative homogenization somewhere here?

The arguments in Chapter LABEL:c.one only need to be slightly modified in order to obtain a more general qualitative homogenization result valid in the case that the unit range of dependence assumption is relaxed to mere ergodicity. In other words, in place of (2.6) we assume instead that

[TABLE]

In fact, the only argument that needs to be modified is the proof of Proposition LABEL:p.nuconv, since it is the only place in the chapter where independence is used. Moreover, that argument is essentially a proof of the subadditive ergodic theorem in the special case of the unit range of dependence assumption (2.6). In the general ergodic case (3.1), one can simply directly apply the subadditive ergodic theorem (see for instance [AK]) to obtain, in place of (LABEL:e.preL1conv), the estimate

[TABLE]

The rest of the arguments in that chapter are deterministic and imply that the random variable $\mathcal{E}^{\prime}(\varepsilon)$ in Theorem LABEL:t.DP.blackbox satisfies $\mathbb{P}\left[\limsup_{\varepsilon\to 0}\mathcal{E}^{\prime}(\varepsilon)=0\right]=1$.

What do we learn about reversible diffusions in random environments?

Just as we learn about Brownian motion from properties of harmonic functions (and conversely), the study of divergence-form operators gives us information about the associated diffusion processes. To start with, De Giorgi-Nash-Aronson estimates recalled in (LABEL:e.upperP.NA) and Proposition LABEL:p.par.reg.NA can be used together with the classical Kolmogorov extension and continuity theorems (see [billingsley, Theorem 36.2] and [revuz-yor, Theorem I.2.1]) to define these stochastic processes. Denoting by $\mathbf{P}_{x}^{\mathbf{a}}$ the probability law of the diffusion process starting from $x\in{\mathbb{R}^{d}}$, and by $(X(t))_{t\geqslant 0}$ the canonical process, we have by construction that, for every $\mathbf{a}\in\Omega$, Borel measurable set $A\subseteq{\mathbb{R}^{d}}$ and $(t,x)\in(0,\infty)\times{\mathbb{R}^{d}}$,

[TABLE]

where $P(t,x,y)$ is the parabolic Green function defined in Proposition LABEL:p.GFexistence.NA. The statement

[TABLE]

where $\overline{P}$ is the parabolic Green function for the homogenized operator, can thus be interpreted as a (quenched) local central limit theorem for the diffusion process. Seen in this light, Theorem LABEL:t.PGF.basecase gives us a first quantitative version of this local central limit theorem. The much more precise Theorem LABEL:t.PtoPbar gives an optimal rate of convergence for this statement, and can thus be interpreted as analogous to the classical Berry-Esseen result on the rate of convergence in the central limit theorem for sums of independent random variables (see [Petrov, Theorem 5.5]).

Chapter 4 Notation

Sets and Euclidean space

The set of nonnegative integers is denoted by $\mathbb{N}:=\{0,1,2,\ldots\}$. The set of real numbers is written $\mathbb{R}$. When we write $\mathbb{R}^{m}$ we implicitly assume that $m\in\mathbb{N}\setminus\{0\}$. For each $x,y\in\mathbb{R}^{m}$, the scalar product of $x$ and $y$ is denoted by $x\cdot y$, their tensor product by $x\otimes y$ and the Euclidean norm on $\mathbb{R}^{m}$ is $\left|\,\cdot\,\right|$. The canonical basis of $\mathbb{R}^{m}$ is written as $\{e_{1},\ldots,e_{m}\}$. We let $\mathcal{B}$ denote the Borel $\sigma$–algebra on $\mathbb{R}^{m}$. A domain is an open connected subset of $\mathbb{R}^{m}$. The notions of $C^{k,\alpha}$ domain and Lipschitz domain are defined in Definition LABEL:def.Lipdomain. The boundary of $U\subseteq\mathbb{R}^{m}$ is denoted by $\partial U$ and its closure by $\overline{U}$. The open ball of radius $r>0$ centered at $x\in\mathbb{R}^{m}$ is $B_{r}(x):=\left\{y\in\mathbb{R}^{m}\,:\,|x-y|<r\right\}$. The distance from a point to a set $V\subseteq\mathbb{R}^{m}$ is written $\operatorname{dist}(x,V):=\inf\left\{|x-y|\,:\,y\in V\right\}$. For $r>0$ and $U\subseteq\mathbb{R}^{m}$, we define

[TABLE]

For $\lambda>0$, we set $\lambda U:=\left\{\lambda x\,:\,x\in U\right\}$. If $m,n\in\mathbb{N}\setminus\{0\}$, the set of $m\times n$ matrices with real entries is denoted by $\mathbb{R}^{m\times n}$. We typically denote an element of $\mathbb{R}^{m\times n}$ by a boldfaced latin letter, such as $\mathbf{m}$, and its entries by $(\mathbf{m}_{ij})$. The subset of $\mathbb{R}^{n\times n}$ of symmetric matrices is written $\mathbb{R}^{n\times n}_{\mathrm{sym}}$ and the set of $n$-by-$n$ skew-symmetric matrices is $\mathbb{R}^{n\times n}_{\mathrm{skew}}$. The identity matrix is denoted $\mathsf{Id}$. If $r,s\in\mathbb{R}$ then we write $r\vee s:=\max\{r,s\}$ and $r\wedge s:=\min\{r,s\}$. We also denote $r_{+}:=r\vee 0$ and $r_{-}:=-(r\wedge 0)$.

We use triadic cubes throughout the book. For each $m\in\mathbb{N}$, we denote

[TABLE]

Observe that, for each $n\in\mathbb{N}$ with $n\leqslant m$, the cube ${\scaleobj{1.2}{\square}}_{m}$ can be partitioned (up to a set of zero Lebesgue measure) into exactly $3^{d(m-n)}$ subcubes which are $\mathbb{Z}^{d}$-translations of ${\scaleobj{1.2}{\square}}_{n}$, namely $\left\{z+{\scaleobj{1.2}{\square}}_{n}\,:\,z\in 3^{n}\mathbb{Z}^{d}\cap{\scaleobj{1.2}{\square}}_{m}\right\}$.

Calculus

If $U\subseteq{\mathbb{R}^{d}}$ and $f:U\to\mathbb{R}$, we denote the partial derivatives of $f$ by $\partial_{x_{i}}f$ or simply $\partial_{i}f$, which unless otherwise indicated, is understood in the sense of distributions. The gradient of $f$ is denoted by $\nabla f:=\left(\partial_{1}f,\ldots,\partial_{d}f\right)$. The Hessian of $f$ is denoted by $\nabla^{2}f:=\left(\partial_{i}\partial_{j}f\right)_{i,j\in\{1,\ldots,d\}}$, and higher derivatives are denoted similarly:

[TABLE]

A vector field on $U\subseteq{\mathbb{R}^{d}}$, typically denoted by a boldfaced latin letter, is a function $\mathbf{f}:U\to{\mathbb{R}^{d}}$. The divergence of $\mathbf{f}$ is $\nabla\cdot\mathbf{f}=\sum_{k=1}^{d}\partial_{i}f_{i}$, where the $(f_{i})$ are the entries of $\mathbf{f}$, i.e., $\mathbf{f}=\left(f_{1},\ldots f_{d}\right)$.

Hölder and Lebesgue spaces

For $k\in\mathbb{N}\cup\{\infty\}$, the set of functions $f:U\to\mathbb{R}$ which are $k$ times continuously differentiable in the classical sense is denoted by $C^{k}(U)$. We denote by $C^{k}_{c}(U)$ the collections of $C^{k}(U)$ functions with compact support in $U$. For $k\in\mathbb{N}$ and $\alpha\in(0,1]$, we denote the classical Hölder spaces by $C^{k,\alpha}(U)$, which are the functions $u\in C^{k}(U)$ for which the norm

[TABLE]

is finite, where $\left[\,\cdot\,\right]_{C^{0,\alpha}(U)}$ is the seminorm defined by

[TABLE]

For every Borel set $U\in\mathcal{B}$, we denote by $|U|$ the Lebesgue measure of $U$. For an integrable function $f:U\to\mathbb{R}$, we may denote the integral of $f$ in a compact notation by

[TABLE]

For $U\subseteq{\mathbb{R}^{d}}$ and $p\in[1,\infty]$, we denote by $L^{p}(U)$ the Lebesgue space on $U$ with exponent $p$, that is, the set of measurable functions $f:U\to\mathbb{R}$ satisfying

[TABLE]

The vector space of functions on $U$ which belong to $L^{p}(V)$ whenever $V$ is bounded and $\overline{V}\subseteq U$ is denoted by $L^{p}_{\mathrm{loc}}(U)$. If $|U|<\infty$ and $f\in L^{1}(U)$, then we write

[TABLE]

The average of a function $f\in L^{1}(U)$ on $U$ is also sometimes denoted by

[TABLE]

To make it easier to keep track of scalings, we very often work with rescaled versions of $L^{p}$ norms: for every $p\in[1,\infty)$ and $f\in L^{p}(U)$, we set

[TABLE]

For convenience, we may also use the notation $\|f\|_{{\underline{L}}^{\infty}(U)}:=\|f\|_{L^{\infty}(U)}$. If $X$ is a Banach space, then $L^{p}(U;X)$ denotes the set of measurable functions $f:U\to X$ such that $x\mapsto\left\|f(x)\right\|_{X}\in L^{p}(U)$. We denote the corresponding norm by $\left\|f\right\|_{L^{p}(U;X)}$. By abuse of notation, we will sometimes write $\mathbf{f}\in L^{p}(U)$ if $\mathbf{f}:U\to\mathbb{R}^{m}$ is a vector field such that $|\mathbf{f}|\in L^{p}(U)$ and denote $\|\mathbf{f}\|_{\underline{L}^{p}(U)}:=\|\mathbf{f}\|_{\underline{L}^{p}(U;\mathbb{R}^{m})}=\||\mathbf{f}|\|_{\underline{L}^{p}(U)}$. For $f\in L^{p}({\mathbb{R}^{d}})$ and $g\in L^{p^{\prime}}({\mathbb{R}^{d}})$ with $\frac{1}{p}+\frac{1}{p^{\prime}}=1$, we denote the convolution of $f$ and $g$ by

[TABLE]

Special functions

For $p\in{\mathbb{R}^{d}}$, we denote the affine function with slope $p$ passing through the origin by

[TABLE]

Unless otherwise indicated, $\zeta\in C^{\infty}_{c}({\mathbb{R}^{d}})$ denotes the standard mollifier

[TABLE]

with the multiplicative constant $c_{d}$ chosen so that $\int_{\mathbb{R}^{d}}\zeta=1$. We denote, for $\delta>0$,

[TABLE]

The standard heat kernel is denoted by

[TABLE]

and define, for each $z\in{\mathbb{R}^{d}}$ and $r>0$,

[TABLE]

We also denote by $\mathcal{P}_{k}$ the set of real polynomials on ${\mathbb{R}^{d}}$ of order at most $k$.

Sobolev and fractional Sobolev spaces

For $k\in\mathbb{N}$ and $p\in[1,\infty]$, we denote by $W^{k,p}(U)$ the classical Sobolev space, see Definition LABEL:def.Sobolev in Appendix LABEL:a.sobolev. The corresponding norm is $\left\|\,\cdot\,\right\|_{W^{k,p}(U)}$. For $\alpha\in(0,\infty)\setminus\mathbb{N}$, the space $W^{\alpha,p}(U)$ is the fractional Sobolev space introduced in Definition LABEL:def.sobolev.fractional, with norm $\left\|\,\cdot\,\right\|_{W^{\alpha,p}(U)}$. For $\alpha\in(0,\infty)$, we denote by $W^{\alpha,p}_{0}(U)$ the closure of $C^{\infty}_{c}(U)$ in $W^{\alpha,p}(U)$. We also use the shorthand notation

[TABLE]

with corresponding norms $\left\|\,\cdot\,\right\|_{H^{\alpha}(U)}=\left\|\,\cdot\,\right\|_{W^{\alpha,p}(U)}$.

As for $L^{p}$ spaces, it is useful to work with normalized and scale-invariant versions of the Sobolev norms. We define the rescaled $W^{k,p}(U)$ norm of a function $u\in W^{k,p}(U)$ by

[TABLE]

Observe that

[TABLE]

We also set $\left\|u\right\|_{\underline{H}^{1}(U)}:=\left\|u\right\|_{\underline{W}^{1,2}(U)}$. We use the notation $W^{\alpha,p}_{\mathrm{loc}}(U)$ and $H^{\alpha}_{\mathrm{loc}}(U)$ for the spaces defined analogously to $L^{p}_{\mathrm{loc}}(U)$.

Negative Sobolev spaces

For $\alpha\in(0,\infty)$, $p\in[1,\infty]$ and $p^{\prime}:=\frac{p}{p-1}$ the Hölder conjugate exponent of $p$, the space $W^{-\alpha,p}(U)$ is the space of distributions $u$ such that the following norm is finite:

[TABLE]

We also set $H^{-\alpha}(U):=W^{-\alpha,2}(U)$. For $p>1$, the space $W^{-\alpha,p}(U)$ is the space dual to $W^{\alpha,p^{\prime}}_{0}(U)$, and we have

[TABLE]

We refer to Definition LABEL:def.negsob and Remark LABEL:r.non.separable.sobolev for details. The rescaled $W^{-\alpha,p}(U)$ norm is defined by

[TABLE]

We also set $\left\|\,\cdot\,\right\|_{\underline{H}^{-1}(U)}=\left\|\,\cdot\,\right\|_{\underline{W}^{-1,2}(U)}$. These rescaled norms behave under dilations in the following way:

[TABLE]

Note that we have abused notation in (4.7)-(4.8), denoting by $\int_{U}uv$ the duality pairing between $u$ and $v$. In other words, “$\int_{U}uv$” denotes the duality pairing that is normalized in such a way that if $u,v\in C^{\infty}_{c}(U)$, then the notation agrees with the usual integral. In (4.9), we understand that $\strokedint_{U}uv=|U|^{-1}\int_{U}uv$.

It is sometimes useful to consider the slightly different space $\widehat{\phantom{H}}\mathchoice{\hbox{}}{\hbox{}}{\hbox{}}{\hbox{}}H^{-1}(U)$ which is the dual space to $H^{1}(U)$. We denote its (rescaled) norm by

[TABLE]

It is evident that $\widehat{\phantom{H}}\mathchoice{\hbox{}}{\hbox{}}{\hbox{}}{\hbox{}}H^{-1}(U)\subseteq H^{-1}(U)$ and that we have

[TABLE]

Solenoidal and potential fields

We let $L^{2}_{\mathrm{pot}}(U)$ and $L^{2}_{\mathrm{sol}}(U)$ denote the closed subspaces of $L^{2}(U;{\mathbb{R}^{d}})$ consisting respectively of potential (gradient) and solenoidal (divergence-free) vector fields. These are defined by

[TABLE]

and

[TABLE]

Here we interpret the condition $\nabla\cdot\mathbf{g}=0$ in the sense of distributions. In this case, it is equivalent to the condition

[TABLE]

We can also make sense of the condition $\nabla\cdot\mathbf{g}=0$ if the entries of the vector field $\mathbf{g}$ are distributions by restricting the condition above to $\phi\in C^{\infty}_{c}(U)$. We also set

[TABLE]

and

[TABLE]

Notice that from these definitions we immediately have the Helmholtz-Hodge orthogonal decompositions

[TABLE]

which is understood with respect to the usual inner product on $L^{2}(U;{\mathbb{R}^{d}})$. Finally, we denote

[TABLE]

and

[TABLE]

The probability space

As explained previously in the assumptions section, except where explicitely indicated to the contrary, $(\Omega,\mathcal{F})$ refers to the pair defined in (2.2) and (2.3) which is endowed by the probability measure $\mathbb{P}$ satisfying the assumptions (2.5) and (2.6). The constants $\Lambda\geqslant 1$ and $d\in\mathbb{N}$ in the definition of $\Omega$ (i.e., the ellipticity constant and the dimension) are fixed throughout the book.

A random element on a measurable space $(G,\mathcal{G})$ is an $\mathcal{F}$–measurable map $X:\Omega\to G$. If $G$ is also a topological space and we do not specify $\mathcal{G}$, it is understood to be the Borel $\sigma$-algebra. In the case that $G=\mathbb{R}$ or $G=\mathbb{R}^{m}$, we say respectively that $X$ is a random variable or vector. If $G$ is a function space on ${\mathbb{R}^{d}}$, then we typically say that $X$ is a random field. If $X$ is a random element taking values in a topological vector space, then we define $\mathbb{E}\left[X\right]$ to be the expectation with respect to $\mathbb{P}$ via the formula (2.7). Any usage of the expression “almost surely,” also abbreviated “a.s.” or “$\mathbb{P}$–a.s.,” is understood to be interpreted with respect to $\mathbb{P}$. For example, if $\{X,X_{1},X_{2},\ldots\}_{n\in\mathbb{N}}$ is a sequence of random variables, then we may write either “$X_{n}\to X$ as $n\to\infty$, $\mathbb{P}$–a.s.,” or

[TABLE]

in place of the statement $\mathbb{P}\left[\limsup_{n\to\infty}\left|X_{n}-X\right|=0\right]=1$.

We denote by $\mathbf{a}$ the canonical element of $\Omega$ which we consider to be a random field by identifying it with the identity map $\mathbf{a}\mapsto\mathbf{a}$. Therefore, all random elements can be considered as functions of $\mathbf{a}$ and conversely, any object which we can define as a function of $\mathbf{a}$ in an $\mathcal{F}$–measurable way. Since this will be the case of nearly every object in the entire book, we typically do not display the dependence on $\mathbf{a}$ explicitly.

We note that this convention is in contrast to the usual one made in the literature on stochastic homogenization, where it is more customary to let $(\Omega,\mathcal{F})$ be an abstract probability space, denote a typical element of $\Omega$ by $\omega$, and consider the coefficient field $\mathbf{a}=\mathbf{a}(x,\omega)$ to be a matrix-valued mapping on ${\mathbb{R}^{d}}\times\Omega$. The $\omega$ then appears everywhere in the notation, since all objects are functions of $\omega$, which in our opinion makes the notation unnecessarily heavy without bringing any additional clarity. We hope that, for readers who find this notation more familiar, it is nevertheless obvious that it is equivalent to ours via the identification of the abstract $\Omega$ with our $\Omega$ by the map $\omega\mapsto\mathbf{a}(\cdot,\omega)$.

Solutions of the PDE

Given a coefficient field $\mathbf{a}\in\Omega$ and an open subset $U\subseteq{\mathbb{R}^{d}}$, the set of weak solutions of the equation

[TABLE]

is denoted by

[TABLE]

We sometimes display the spatial dependence by writing

[TABLE]

This is to distinguish it from the equation we obtain by introducing a small parameter $\varepsilon>0$ and rescaling, which we also consider, namely

[TABLE]

Notice that these are random equations and $\mathcal{A}(U)$ is a random linear vector space since of course it depends on the coefficient field $\mathbf{a}$. Throughout the book, 

$\mathbf{a}$

  is the homogenized matrix defined in Definition LABEL:d.ahom. We denote the set of solutions of the homogenized equation in $U$ by

[TABLE]

For each $k\in\mathbb{N}$, we denote by $\overline{\mathcal{A}}_{k}$ and $\mathcal{A}_{k}$ the subspaces of $\overline{\mathcal{A}}({\mathbb{R}^{d}})$ and $\mathcal{A}({\mathbb{R}^{d}})$, respectively, which grow like $o\left(|x|^{k+1}\right)$ as measured in the $L^{2}$ norm:

[TABLE]

and

[TABLE]

The $\mathcal{O}_{s}(\cdot)$ notation

Throughout the book, we use the following notation to control the size of random variables: for every random variable $X$ and $s,\theta\in(0,\infty)$, we write

[TABLE]

to denote the statement

[TABLE]

We likewise write

[TABLE]

and

[TABLE]

This notation allows us to write bounds for random variables conveniently, in the spirit of the standard “big-$O$” notation it evokes, enabling us to compress many computations that would otherwise take many lines to write. The basic properties of this notation are collected in Appendix LABEL:a.bigO.

Convention for the constants $C$

Throughout, the symbols $c$ and $C$ denote positive constants which may vary from line to line, or even between multiple occurrences in the same line, provided the dependence of these constants is clear from the context. Usually, we use $C$ for large constants (those we expect to belong to $[1,\infty)$) and $c$ for small constants (those we expect to belong to $(0,1]$). When we wish to explicitly declare the dependence of $C$ and $c$ on the various parameters, we do so using parentheses $(\cdots)$. For example, the phrase “there exists $C(p,d,\Lambda)<\infty$ such that…” is short for “there exists a constant $C\in[1,\infty)$, depending on $p$, $d$ and $\Lambda$, such that…” We use the same convention for other symbols when it does not cause confusion, for instance the small positive exponents $\alpha$ appearing in Chapter LABEL:c.two.