An Introduction to the Qualitative and Quantitative Theory

An Introduction to the Qualitative

and Quantitative Theory of Homogenization

Stefan NEUKAMM�

Faculty of Mathematics, Technische Universitat Dresden

We present an introduction to periodic and stochastic homogenization of elliptic partial differential equations.The first part is concerned with the qualitative theory, which we present for equations with periodic and randomcoefficients in a unified approach based on Tartar’s method of oscillating test functions. In particular, we present aself-contained and elementary argument for the construction of the sublinear corrector of stochastichomogenization. (The argument also applies to elliptic systems and in particular to linear elasticity). In thesecond part we briefly discuss the representation of the homogenization error by means of a two-scale expansion.In the last part we discuss some results of quantitative stochastic homogenization in a discrete setting. In particular,we discuss the quantification of ergodicity via concentration inequalities, and we illustrate that the latter incombination with elliptic regularity theory leads to a quantification of the growth of the sublinear corrector and thehomogenization error.

KEYWORDS: stochastic homogenization, quantitative stochastic homogenization, corrector, two-scaleexpansion

Preface and Acknowledgments The present notes originate from a one week mini-course given by the author duringthe GSIS International Winter School 2017 on ‘‘Stochastic Homogenization and its Applications’’ at the TohokuUniversity, Sendai, Japan. The author would like to thank the organizers of that workshop, especially Reika Fukuizumi,Jun Masamune and Shigeru Sakaguchi for their very kind hospitality. The present notes are devoted to graduatestudents and young researchers with a basic knowledge in PDE theory and functional analysis. The first three chaptersare rather self-contained and offer an introduction to the basic theory of periodic homogenization and its extension tohomogenization of elliptic operators with random coefficients. The last chapter, which is in parts based on an extendedpreprint to the paper [13] by Antoine Gloria, Felix Otto and the author, is a bit more advanced, since it invokes someinput from elliptic regularity theory (in a discrete setting) that we do not develop in this manuscript. The author wouldlike to thank Mathias Schaffner and Helmer Hoppe for proofreading the original manuscript, and Andreas Kunze forproviding the illustrations and numerical results, which were obtained in his master thesis [23]. The author wassupported by the DFG in the context of TU Dresden’s Institutional Strategy ‘‘The Synergetic University’’.

Contents

1 Introduction — A One-dimensional Example 22 Qualitative Homogenization of Elliptic Equations 5

2.1 Periodic homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Stochastic homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Proof of Lemma 2.16, Lemma 2.22, and Lemma 2.23 . . . . . . . . . . . . . . . . . . . . 263 Two-scale Expansion and Homogenization Error 284 Quantitative Stochastic Homogenization 31

4.1 The discrete framework and the discrete corrector . . . . . . . . . . . . . . . . . . . . . . . . 314.2 Quantification of ergodicity via Spectral Gap . . . . . . . . . . . . . . . . . . . . . . . . . . 344.3 Quantification of sublinearity in dimension d � 2 . . . . . . . . . . . . . . . . . . . . . . . . 38

A Solutions to Problem 1–5 46

�Corresponding author. E-mail: [email protected]

Received July 27, 2017; Accepted October 31, 2017

Interdisciplinary Information Sciences Vol. 24, No. 1 (2018) 1–48#Graduate School of Information Sciences, Tohoku UniversityISSN 1340-9050 print/1347-6157 onlineDOI 10.4036/iis.2018.A.01

http://dx.doi.org/10.4036/iis.2018.A.01

1. Introduction — A One-dimensional Example

Consider a heat conducting body that occupies some domain O � Rd , where d ¼ 1; 2; . . . denotes the dimension.Suppose that the body is exposed to a heat source/sink that does not vary in time, and suppose that the body is cooled atits boundary, such that its temperature is zero at the boundary. If time evolves the temperature of the body willconverge to a steady state, which can be described by the elliptic boundary value problem

�r � ðaruÞ ¼ f in O;

u ¼ 0 on @O:

In this equation. u : O! R denotes the (sought for) temperature field,. f : O! R is given and describes the heat source.

The ability of the material to conduct heat is described by a material parameter a 2 ð0;1Þ, called the conductivity. Thematerial is homogeneous, if a does not depend on x. The material is called heterogeneous, if aðxÞ varies in x 2 O. In thislecture we are interested in heterogeneous materials with microstructure, which means that the heterogeneity varies ona length scale, called the microscale, that is much smaller than a macroscopic length scale of the problem, e.g., thediameter of the domain O or the length scale of the right-hand side f .

To fix ideas, suppose that aðxÞ ¼ a0ðx‘Þ with a0 periodic, i.e., the conductivity is periodic with the period ‘. If the ratio

" :¼ microscalemacroscale

¼ ‘L

is a small number, e.g., " . 10�3, then we are in the regime of a microstructured material. The goal of homogenizationis to derive a simplified PDE by studying the limit " # 0, i.e., when the micro- and macroscale separate.

In the rest of the introduction we treat the following one-dimensional example: Let O ¼ ð0;LÞ � R, " > 0 and letu" : O! R be a solution to the equation

�@xðaðx"Þ@xu"ðxÞÞ ¼ f in O; ð1:1Þu" ¼ 0 on @O: ð1:2Þ

We suppose that a : R! R is 1-periodic and uniformly elliptic, i.e., there exists � > 0 such that aðxÞ 2 ð�; 1Þ for allx 2 R. For simplicity, we assume that f and a are smooth.

We are going to prove the following homogenization result:. For all " > 0 equations ð1.1Þ, ð1.2Þ admit a unique smooth solution u".. As " # 0, u" converges to a smooth function u0.. The limit u0 is the unique solution to the equation

�@xða0@xu0Þ ¼ f in O; ð1:3Þu0 ¼ 0 on @O; ð1:4Þ

where a0 2 R denotes the harmonic mean of a, i.e.,

a0 ¼Z 1

0

a�1ðyÞ dy� ��1

:

Problem 1. Show that ð1.1Þ and ð1.2Þ admit a unique, smooth solution.

The solution to this and all subsequent problems in this introduction can be found in Appendix A. We have anexplicit presentation for the solution:

u"ðxÞ ¼Z x

0

a�1" ðx

0Þ c" �Z x0

0

f ðx00Þ dx00� �

dx0; ð1:5Þ

where

c" ¼Z L

0

a�1" ðx

0Þ dx0� ��1Z L

0

Z x0

0

a�1" ðx

0Þ f ðx00Þ dx00 dx0:

In order to pass to the limit " # 0 in the representation ð1.5Þ, we need to understand the limit of functions of the form

x 7!1

aðx"Þ

Z x

0

f ðx0Þ dx0:

This function rapidly oscillates on scale " and the amplitude of the oscillations is of unit order. Hence, the expressiondoes not converge uniformly (or in a strong sense). Nevertheless, we have the following result:

2 NEUKAMM

Lemma 1.1. Let Fðy; xÞ be a smooth function that is 1-periodic in y 2 R and assume that F and @xF are bounded.Then

lim"#0

Z b

a

Fðx"; xÞ dx ¼Z b

a

�FðxÞ dx; �FðxÞ ¼Z 1

0

Fðy; xÞ dy:

Furthermore, there exists a constant C (only depending on F) such thatZ b

a

ðFðx"; xÞ � �FðxÞÞ dx��

�� Cðjb� aj þ 1Þ":

Proof. Consider the functions

Gðy; xÞ ¼Z y

0

ðFðy0; xÞ � �FðxÞÞ dy0; g"ðxÞ :¼ "Gðx"; xÞ:

Note that Gðy; xÞ and @xGðy; xÞ are 1-periodic in y; indeed, we have

Gðyþ 1; xÞ � Gðy; xÞ ¼Z yþ1

y

ðFðy0; xÞ � �FðxÞÞ dy0 ¼ �FðxÞ � �FðxÞ ¼ 0;

and the same is true for @xG. Furthermore, G and @xG are smooth and bounded, and we have

@xg"ðxÞ ¼ "@xGðx"; xÞ þ @yGðx"; xÞ ¼ "@xGðx"; xÞ þ ðFð

x"; xÞ � �FðxÞÞ;

and thus Z b

a

ðFðx"; xÞ � �FðxÞÞ dx ¼

Z b

a

ð@xg"ðxÞ � "@xGðx; x"ÞÞ dx

¼ " Gðb"; bÞ � Gða

"; aÞ �

Z b

a

@xGðx"; xÞ dx� �

:

(a) (b) (c)

(d) (e) (f)

Fig. 1. (a)–(c) show the rapidly oscillating coefficient field aðx"Þ ¼ 2þ sinð2� x"Þ for " 2 f1

4; 1

16; 1

32g. (d)–(f) show the solutions to

ð1.1Þ and ð1.2Þ with f ðxÞ ¼ �3ð2x� 1Þ.

An Introduction to the Qualitative and Quantitative Theory of Homogenization 3

The expression in the brackets is bounded uniformly in " (by smoothness and periodicity of G and @xG), and thus thestatement follows. �

Problem 2. Show that maxx2O ju"ðxÞ � u0ðxÞj � C" where C only depends on O, f and a.

The physical interpretation of the result of Problem 2 is the following: While the initial problem ð1.1Þ & ð1.2Þdescribes a heterogeneous, microstructured material (a periodic composite with period "), the limiting equation ð1.3Þ &ð1.4Þ describes a homogeneous material with conductivity a0. Hence, Problem 2 states that if we observe a materialwith a rapidly oscillating conductivity að�"Þ on a macroscopic length scale, then it behaves like a homogeneous materialwith effective conductivity given by a0. We therefore call ð1.3Þ & ð1.4Þ the homogenized problem. It is much simplerthan the heterogeneous initial problem ð1.1Þ & ð1.2Þ:

Problem 3. Let f � 1. Show that a solution to

�@xða@xuÞ ¼ 1 in O;

u ¼ 0 on @O:

is a quadratic function, if and only if the material is homogeneous, i.e., iff a does not depend on x.

The homogenization result shows that u"! u0 as " # 0. Hence, for " 1 the function u0 is a consistentapproximation to the solution to ð1.1Þ & ð1.2Þ. We even have a rate: u" ¼ u0 þ Oð"Þ. Thanks to the homogenizationresult certain properties of the difficult equation ð1.1Þ & ð1.2Þ can be studied by analyzing the simpler problem ð1.3Þ &ð1.4Þ:

Problem 4. Let f � 1 and O ¼ ð0; 1Þ. Show that M" :¼ max �O u" ¼ 18a0þ Oð"Þ.

What can be said about the convergence of the gradient @xu"?

Problem 5. Show that lim supROj@xu" � @xu0j2 > 0 (unless the initial material is homogeneous). Show on the other

hand, that for all smooth functions ’ : R! R we haveZO

u0"ðxÞ’ðxÞ dx!ZO

u00ðxÞ’ðxÞ dx

i.e., we have weak convergence, but not strong convergence.

Yet, we can modify u0 by adding oscillations, such that the gradient of the modified functions converges:

Lemma 1.2 (Two-scale expansion). Let a; f be smooth, O ¼ ð0; 1Þ. Let � : R! R denote a 1-periodic solution to

@yðaðyÞð@y�ðyÞ þ 1ÞÞ ¼ 0 ð1:6Þ

with �ð0Þ ¼ 0. Let u0 and u" be as above. Consider

v"ðxÞ :¼ u0ðxÞ þ "�ðx"Þ@xu0ðxÞ:

Then there exists a constant C > 0 such that for all " > 0 with 1" 2 N we haveZ

O

ju" � v"j2 þ j@xu" � @xv"j2 �4

�2max j�j2

� �"2ZO

j@2xu0j2:

Proof. To ease notation we write

a"ðxÞ :¼ aðx"Þ; �"ðxÞ :¼ �ðx"Þ:

Step 1.It can be easily checked (by direct calculations) that

�ðyÞ :¼Z y

0

a0

aðtÞ� 1

� �dt

and that � is smooth and bounded. Note that

a0 ¼ aðyÞð@y�ðyÞ þ 1Þ for all y 2 R:

Indeed, by the corrector equation ð1.6Þ and the definition of a0 the difference of both functions is constant and has zeromean. (This is only true in the one-dimensional case!)Step 2.Set z" :¼ u" � v". Since 1

" 2 N we have �ð1"Þ ¼ 0. Combined with the boundary conditions imposed on u" and �" weconclude that z"ð0Þ ¼ z"ð1Þ ¼ 0. We claim that

4 NEUKAMM

ZO

jz"j2 �ZO

j@xz"j2:

Indeed, since O ¼ ð0; 1Þ and z" ¼ 0 on @O, this follows by Poincare’s inequality:Z 1

0

jz"j2 ¼Z 1

0

Z x

0

@xz"

� �2

�Z 1

0

j@xz"j2:

Hence, ZO

jz"j2 þ j@xz"j2 � 2

ZO

j@xz"j2 � 2�

ZO

j@xz"j2a";

where we used that a" � � by assumption. Since z" ¼ 0 on @O, we may integrate by parts and getZO

jz"j2 þ j@xz"j2 � 2�

ZO

z"ð�@xða"@xz"ÞÞ:

Step 3. We compute ð�@xða"@xz"ÞÞ:

@xz" ¼ @xu" � ð@y�ðx"Þ þ 1Þ@xu0 � "�"@2xu0

use a0 ¼ a"ð@y�ð�"Þ þ 1Þa"@xz" ¼ a"@xu" � a0@xu0 � "a"�"@2xu0

�@xða"@xz"Þ ¼ �@xða"@xu"Þ þ @xða0@xu0Þ þ @xð"a"�"@2xu0Þ:

The first two terms on the right-hand side are equal to the left-hand side of the PDEs for u" and u0. Hence, these twoterms evaluate to f � f ¼ 0:

�@xða"@xz"Þ ¼ @xð"a"�"@2xu0Þ:

Combined with the estimate of Step 2 we deduce thatZO

jz"j2 þ j@xz"j2 � 2�

ZO

z"@xð"a"�"@2xu0Þ

integration by parts

¼ �2

�

ZO

@xz"ð"�"a"@2xu0Þ

Cauchy{Schwarz and Young’s inequality

in the form ab � �2a2 þ 1

2� b2 with � ¼ �

2

� 12

ZO

j@xz"j2 þ 2�2"

2

ZO

j�"j2ja"j2j@2xu0j2;

and thus ZO

jz"j2 þ j@xz"j2 �4

�2"2ZO

j�"j2j@2xu0j2:

�

In this lecture we extend the previous one-dimensional results to. higher dimensions — the argument presented above heavily relies on the fact that we have an explicit

representation for the solutions. In higher dimensions such a representation is not available and the argument willbe more involved. In particular, we require some input from the theory of partial differential equations andfunctional analysis such as the notion of distributional solutions, the existence theory for elliptic equations indivergence form in Sobolev spaces, the theorem of Lax–Milgram, Poincare’s inequality, the notion of weakconvergence in L2-spaces, and the theorem of Rellich–Kondrachov, e.g., see the textbook on functional analysisby Brezis [8].

. periodic and random coefficients — to treat the later we require some input from ergodic & probability theory.Moreover, we discuss

. the two-scale expansion in higher dimension and in the stochastic case, and explain

. quantitative results for stochastic homogenization in a discrete setting.

2. Qualitative Homogenization of Elliptic Equations

In this section we discuss the homogenization theory for elliptic operators of the form �r � ðarÞ with uniformlyelliptic coefficients. We say that a : Rd ! Rdd is uniformly elliptic with ellipticity constant � > 0, and writea 2 MðRd ; �Þ, if a is measurable, and for a.e. x 2 Rd we have


8� 2 Rd : � � aðxÞ� � �j�j2 and jaðxÞ�j � j�j: ð2:1Þ

A standard result (that invokes the Lax–Milgram Theorem) yields existence of weak solutions to the associated ellipticboundary value problem.

Problem 6. Let a 2 MðRd; �Þ, O � Rd open and bounded, f 2 L2ðOÞ, F 2 L2ðO;RdÞ. Show that there exists a uniquesolution u 2 H1

0ðOÞ to the equation

�r � ðaruÞ ¼ f � r � F in D0ðOÞ: ð2:2Þ

It satisfies the a priori estimate

kukH1ðOÞ � Cð�; d; diamðOÞÞðk fkL2ðOÞ þ kFkL2ðOÞÞ: ð2:3Þ

In this section we study a classical problem of elliptic homogenization: Given a family of coefficient fields ða"Þ �MðRd; �Þ, consider the weak solution u" 2 H1

0ðOÞ to the equation �r � ða"ru"Þ ¼ f �r � F in D0ðOÞ. A prototypicalhomogenization result states that under appropriate conditions on ða"Þ,

. u" weakly converges to a limit u0 in H10 ðOÞ as " # 0.

. The limit u0 can be characterized as the unique weak solution in H10ðOÞ to a homogenized equation

�r � ðahomru0Þ ¼ f � r � F.. The homogenized coefficient field ahom can be computed from ða"Þ by a homogenization formula.

We discuss two types of structural conditions on the coefficient fields ða"Þ that allow to prove such a result. In the firstcase, which usually is referred to as periodic homogenization, the coefficient fields are assumed to be periodic, i.e.,a"ð�Þ ¼ a0ð �"Þ, where a0 is periodic in the following sense:

Definition 2.1. We call a measurable function f defined on Rd L-periodic, if for all z 2 Zd we have

f ð� þ LzÞ ¼ f ð�Þ a.e. in Rd:

In the second case, called stochastic homogenization, the coefficient fields are supposed to be stationary and ergodicrandom coefficients. We discuss the stochastic case in more detail in Sect. 2.2.

Both cases (the periodic and the stochastic case) can be analyzed by a common approach that relies on Tartar’smethod of oscillating test function, see [26]. In the following we present the approach in the periodic case in a form thateasily adapts to the stochastic case.

2.1 Periodic homogenization

In this section we prove the following classical and prototypical result of periodic homogenization.

Theorem 2.2 (e.g., see textbook Bensoussan, Lions and Papanicolaou [6]). Let � > 0 and a 2 MðRd ; �Þ be 1-periodic. Then there exists a constant, uniformly elliptic coefficient matrix ahom such that:For all O � Rd open and bounded, for all f 2 L2ðOÞ and F 2 L2ðO;RdÞ, and " > 0, the unique weak solutionu" 2 H1

0ðOÞ to

�r � ðaðx"Þru"Þ ¼ f �r � F in D0ðOÞ

weakly converges in H1ðOÞ to the unique weak solution u0 2 H10ðOÞ to

�r � ðahomru0Þ ¼ f �r � F in D0ðOÞ:

Above and throughout the paper we write �r � F ¼ f in D0ðOÞ to express that the identity holds in the distributionalsense, i.e.,

ROF � r’ ¼

ROf’ for all ’ 2 C1c ðOÞ. A numerical illustration of the theorem is depicted in Fig. 2.

The main difficulty in the proof of the theorem is to pass to the limit in expressions of the formZO

aðx"Þru"ðxÞ � �ðxÞei dx ð� 2 C1c ðOÞÞ;

since the integrand is a product of weakly convergent terms. In a nutshell Tartar’s method relies on the idea toapproximate the test field �ðxÞei by some gradient field rð�gi;"Þ, where gi;" denotes an oscillating test function with theproperty that �r � atð�"Þrgi;" !�r � a

thomei in H�1ðOÞ. We can then pass to the limit by appealing to the following

special form of Murat & Tartar’s celebrated div-curl lemma, see [26]:

Lemma 2.3. Consider ðu"Þ � H10ðOÞ and ðF"Þ � L2ðO;RdÞ. Suppose that

. u" * u0 weakly in H10 ðOÞ,

. F" * F0 weakly in L2ðO;RdÞ andROF" � r�"!

ROF � r� for any sequence ð�"Þ � H1

0ðOÞ with �" * � weakly inH1ðOÞ.

Then for any � 2 C1c ðOÞ we have

6 NEUKAMM

ZO

�ðru" � F"Þ !ZO

�ðru0 � F0Þ:

Proof. ZO

�ðru" � F"Þ ¼ZO

rðu"�Þ � F" �ZO

u"r� � F":

Since u"� * u0� weakly in H10 ðOÞ, and u"r�! u0r� strongly in L2ðOÞ (by the Rellich–Kondrachov Theorem), we

find that the right-hand side converges toZO

rðu0�Þ � F0 �ZO

u0r� � F0 ¼ZO

�ðru0 � F0Þ:

�

It turns out that the homogenization result holds, whenever we are able to construct an oscillating test function gi;". Thismotivates the following definition:

Definition 2.4. We say that ða"Þ � MðRd; �Þ admits homogenization if there exists an elliptic, constant coefficientmatrix ahom, called ‘‘the homogenized coefficients’’, such that the following properties hold: For i ¼ 1; . . . ; d there existoscillating test functions ðgi;"Þ � H1

locðRdÞ such that

�r � at"rgi;" ¼ 0 in D0ðRdÞ; ðC1Þgi;" * xi weakly in H1

locðRdÞ; ðC2Þ

at"rgi;" * athomei weakly in L2locðR

dÞ: ðC3Þ

(a) (b)

(c) (d)

Fig. 2. Illustration of Theorem 2.2 in the periodic, two-dimensional case. (a) shows a periodic checkerboard-like coefficient fieldað�Þ. (b)–(c) show the solution to the equation �r � ðað�"Þru"Þ ¼ 1 on the unit cube with homogeneous Dirichlet boundary valuesfor " 2 f1

2; 1

8; 1

32g.


Based on (C1)–(C3) and the div-curl lemma we obtain the following general homogenization result:

Lemma 2.5. Suppose ða"Þ � MðRd; �Þ admits homogenization with homogenized coefficients ahom. Then for allO � Rd open and bounded, for all f 2 L2ðOÞ and F 2 L2ðO;RdÞ, and " > 0, the unique weak solution u" 2 H1

0ðOÞ to

�r � ða"ru"Þ ¼ f � r � F in D0ðOÞ

weakly converges in H1ðOÞ to the unique weak solution u0 2 H10ðOÞ to


Moreover, we have

a"ru" * ahomru0 weakly in L2ðO;RdÞ:

Proof. Step 1. Compactness.We denote the flux by

j" :¼ a"ru":

By the a priori estimates of Problem 6 we haveZO

ju"j2 þ jru"j2 þ jj"j2 � C

ZO

jf j2 þ jFj2

where C does not depend on ". Since bounded sets in L2ðO;RdÞ and H1ðOÞ are precompact in the weak topology, andsince H1ðOÞ b L2

locðOÞ is compactly embedded (by the Rellich–Kondrachov Theorem), there exist u0 2 H10ðOÞ and

j0 2 L2ðOÞ such that, for a subsequence (that we do not relabel), we have

u" * u0 weakly in H1ðOÞ;u"! u0 in L2

locðOÞ;j" * j0 weakly in L2ðOÞ:

We claim that

�r � j0 ¼ f �r � F in D0ðOÞ: ð2:4Þ

Indeed, for all ’ 2 C1c ðOÞ we haveZO

j0 � r’ ZO

j" � r’ ¼ZO

a"ru" � r’ ¼ZO

f � ’þ F � r’:

Step 2. Identification of j0.We first argue that it suffices to prove the identity

j0 ¼ ahomru0: ð2:5Þ

Indeed, the combination of ð2.5Þ and ð2.4Þ shows that


Since this equation has a unique solution (recall that ahom is assumed to be elliptic), we deduce that u0 and j0 (whichwere originally obtained as a weak limits of ðu"Þ and ð j"Þ along a subsequence), are independent of the subsequence.Hence, we get u" * u0 weakly in H1ðOÞ and j" * ahomru0 weakly in L2ðO;RdÞ along the entire sequence, and thus theclaimed statement follows.

It remains to prove ð2.5Þ. By the fundamental lemma of the calculus of variations, it suffices to show: For all� 2 C1c ðOÞ and i ¼ 1; . . . ; d we have Z

O

�ð j0 � ahomru0Þ � ei ¼ 0: ð2:6Þ

For the argument let gi;" denote the oscillating test function of Definition 2.4, and note that for any sequence �" * �0

weakly in H10ðOÞ we have Z

O

j" � r�" ¼ZO

f�" þ F � r�"!ZO

f�0 þ F � �0 ¼ZO

j � r�0:

Hence, an application of the div-curl lemma, see Lemma 2.3, and property (C2) yieldZO

�ð j" � rgi;"Þ !ZO

�ð j0 � eiÞ:

On the other hand, by (C1), (C3) and the convergence of u", the div-curl lemma also yields

8 NEUKAMM

ZQ

�ð j" � rgi;"Þ ¼ZQ

�ðru" � at"rgi;"Þ !ZQ

�ðru0 � athomeiÞ;

and thus ð2.6Þ. �

With Lemma 2.5 at hand, the proof of Theorem 2.2 reduces to the construction of the oscillating test functions gi;".In the periodic case the construction is based on the notion of the periodic corrector. Before we come to its definitionwe introduce a Sobolev space of periodic functions: Let � :¼ ð� 1

2; 1

2Þd denote the unit box in Rd. For L > 0 set

H1#ðL�Þ :¼ fu 2 H1

locðRdÞ : u is L-periodic:g:

Problem 7. Show that. H1

#ðL�Þ with the inner product of H1ðL�Þ is a Hilbert space (and can be identified with a closed linear subspaceof H1ðL�Þ).

. The space of smooth, L-periodic functions on Rd is dense in H1#ðL�Þ.

. For any F 2 H1#ðL�;R

dÞ we have the integration by parts formulaZLðzþ�Þ

r � F ¼ 0 for all L 2 N and z 2 Rd:

Lemma 2.6 (Periodic corrector). Let a 2 MðRd; �Þ be 1-periodic.(a) For i ¼ 1; . . . ; d there exists a unique �i 2 H1

# ð�Þ with>��i ¼ 0 s.t.?

�

aðr�i þ eiÞ � r� ¼ 0 for all � 2 H1#ð�Þ: ð2:7Þ

(b) �i can be characterized as the unique, 1-periodic function �i 2 H1locðR

dÞ with>��i ¼ 0 and

� r � ðaðr�i þ eiÞÞ ¼ 0 in D0ðRdÞ: ð2:8Þ

Proof of Lemma 2.6 part (a). Throughout the proof ‘ denotes a non-negative integer. We set �‘ :¼ ð� 2‘þ12; 2‘þ1

2Þd for

‘ 2 N0 and note that

�‘ ¼[

x2Zd\�‘

ðxþ�Þ up to a null-set;

where the union on the right-hand side is disjoint. We first remark that the problemZ�‘

aðr�‘ þ eiÞ � r� ¼ 0 for all � 2 H1#ð�‘Þ; ð2:9Þ

admits a unique solution �‘ 2 H1# ð�‘Þ satisfying

R�‘�‘ ¼ 0, as follows from the Lax–Milgram theorem and Poincare’s

inequality. In particular, for ‘ ¼ 0 this proves (a). We claim that �‘ ¼ �0 for any ‘ 2 N. Indeed, let � denote a testfunction in H1

#ð�‘Þ. Then by 1-periodicity of aðr�0 þ eiÞ we haveZ�‘

aðr�0 þ eiÞ � r� ¼X

x2Zd\�‘

Zxþ�

aðr�0 þ eiÞ � r�

¼X

x2Zd\�‘

Z�

aðr�0 þ eiÞ � r�ð� þ xÞ

¼Z�

aðr�0 þ eiÞ � r ~�;

where ~� :¼P

x2Zd\�‘ �ð� þ xÞ. By construction we have ~� 2 H1#ð�Þ, and thus the right-hand side is zero (by appealing

to the equation for �0). Hence, we deduce that �0 solves ð2.9Þ and the conditionR�‘�0 ¼ 0. Since ð2.9Þ admits a unique

solution, we deduce that �0 ¼ �‘.We are now in position to prove the equivalence of the problems ð2.7Þ and ð2.8Þ. For the direction ‘‘)’’ it suffices to

show that for arbitrary � 2 C1c ðRdÞ we have Z

aðr�0 þ eiÞ � r� ¼ 0:

Note that here and throughout the paper we simply writeR

instead ofRRd . For the argument, choose ‘ 2 N sufficiently

large such that � ¼ 0 outside �‘. Then � can be extended to a periodic function �‘ 2 H1#ð�‘Þ, and we conclude that

(since �0 ¼ �‘), Zaðr�0 þ eiÞ � r� ¼

Z�‘

aðr�‘ þ eiÞ � r�‘ ¼ 0:


For the other direction let � denote the solution to ð2.8Þ. It suffices to show that for arbitrary � 2 H1#ð�Þ we have?

�

aðr�þ eiÞ � r� ¼ 0: ð2:10Þ

By periodicity, we have for any �‘ 2 C1c ð�‘Þ,?�

aðr�þ eiÞ � r� ¼?�‘

aðr�þ eiÞ � r�

¼?�‘

aðr�þ eiÞ � rð��‘Þ þ?�‘

aðr�þ eiÞ � ðr�ð1� �‘Þ � �r�‘Þ

¼?�‘

aðr�þ eiÞ � ðr�ð1� �‘Þ � �r�‘Þ;

where the last identity holds thanks to ð2.8Þ. Since distð�‘�1;Rd n�‘Þ ¼ 1, we can find a cut-off function �‘ 2 C1c ð�‘Þ

such that 0 � �‘ � 1, �‘ ¼ 1 on �‘�1 and jr�‘j � C with C independent of ‘. We thus conclude that?�‘

aðr�þ eiÞ � ðr�ð1� �‘Þ � �r�‘Þ��

�� ðC þ 1Þ

j�‘ n�‘�1jj�‘j

?�‘n�‘�1

jaðr�þ eiÞjðjr�j þ j�jÞ

¼ ðC þ 1Þj�‘ n�‘�1jj�‘j

?�

jaðr�þ eiÞjðjr�j þ j�jÞ;

where the last identity holds by 1-periodicity of the integrand. In the limit ‘!1, the right-hand side converges to 0,and thus ð2.10Þ follows. �

Definition 2.7 (Periodic corrector and homogenized coefficient). Let a 2 MðRd; �Þ be 1-periodic. The solution �i toð2.7Þ is called the (periodic) corrector in direction ei (associated with a). The matrix ahom 2 Rdd defined by

ahomei :¼?�

aðr�i þ eiÞ ði ¼ 1; . . . ; dÞ

is called the homogenized coefficient (associated with a).

Lemma 2.8 (Properties of the homogenized coefficients). Let a 2 MðRd; �Þ be 1-periodic and denote by ahom theassociated homogenized coefficients.

(a) (ellipticity). For any � 2 Rd we have

� � ahom� � �j�j2:

(b) (invariance under transposition). Let �ti denote the corrector associated with the transposed matrix at. Then

ðahomÞtei ¼?�

atðr�ti þ eiÞ:

(c) (symmetry). If a is symmetric (a.e. in Rd), then ahom is symmetric.

Proof. For � 2 Rd set �� :¼ �i�i and note that �� is the unique solution in H1#ð�Þ with

>�� ¼ 0 to?

�

aðr�� þ �Þ � r� ¼ 0 for all � 2 H1#ð�Þ:

Hence,

1

�� ahom� ¼

1

�

?�

ð�þ r��Þ � að�þ r��Þ �?�

j�þ r��j2 ¼ j�j2 þ?�

jr��j2;

where we used that>�r�� ¼ 0 by periodicity. This proves the ellipticity of ahom. For (b) note that

ðahomÞtei � � ¼ ei � ahom� ¼ ei �?�

aðr�� þ �Þ ¼?�

ðr�ti þ eiÞ � aðr�� þ �Þ

¼?�

atðr�ti þ eiÞ � ðr�� þ �Þ ¼?�

atðr�ti þ eiÞ � �:

Since this is true for arbitrary � 2 Rd, (b) follows. If a is symmetric, then �ti ¼ �i, and thus atðr�ti þ eiÞ ¼ aðr�i þ eiÞ.In this case (b) simplifies to

ðahomÞtei ¼?�

aðr�i þ eiÞ ¼ ahomei;

10 NEUKAMM

and thus ahom is symmetric. �

We finally give the construction of the oscillating test function and establish the properties (C1)–(C3):

Lemma 2.9 (Construction of the oscillating test function). Let a 2 MðRd; �Þ be 1-periodic, let ahom denote theassociated homogenized coefficient and denote by �t1; . . .�

td the periodic correctors associated with the transposed

matrix at. Then ðað�"ÞÞ admits homogenization with homogenized coefficients ahom, and the oscillating test function canbe defined as

gi;"ðxÞ :¼ xi þ "�iðx"Þ:

For the proof we need to pass to the limit in sequences of rapidly oscillating functions:

Proposition 2.10 (Rapidly oscillating functions). Let g 2 L2locðR

dÞ be 1-periodic. Consider g"ðxÞ :¼ gðx"Þ, " > 0. Then

g" * �g :¼?�

g weakly in L2ðOÞ;

for any O � Rd open and bounded.

Proof. Fix O � Rd open and bounded. It suffices to prove:

lim sup"#0kg"k2L2ðOÞ � diamðOÞd

Z�

jgj2; ð2:11Þ

8Q � Rd cube :

?Q

g"! �g: ð2:12Þ

Indeed, this is sufficient, since ð2.11Þ yields boundedness of ðg"Þ, and for weak convergence in L2ðOÞ it suffices to testwith a class of test functions that is dense in L2ðOÞ, e.g., D :¼ spanf1Q indicator function of a cube Q � Og. Now, bylinearity of the integral and by ð2.12Þ, we haveZ

O

g"v!ZO

�gv for all v 2 D:

Step 1. Argument for ð2.11ÞW.l.o.g. let O be a cube. Set Qz;" :¼ zþ "�, and set Z" :¼ fz 2 "Zd : Qz;" \ O 6¼ ;g. Then

kg"k2L2ðOÞ �Xz2Z"

ZQz;"

gx

"

� ��2 dx|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}

¼"dR�

jgj2

¼ "d#Z"kgk2L2ð�Þ;

and the conclusion follows since "d#Z"! jOj � diamðOÞd.Step 2. Argument for ð2.12ÞLet Q denote a cube, set Z" :¼ fz 2 "Zd : Qz;" � Qg, and Q" :¼ [z2Z"Qz;", so that jQ"j ! jQj. ThenZ

Q

g" �ZQ"

g"

��

ZQnQ"jg"j � jQ n Q"j

12kg"kL2ðQÞ ! 0;

andRQz;"

g" ¼ "dR�g, and thus Z

Q"

g" ¼Xz2Z"

ZQz;"

g" ¼ "d#Z"Z�

g ¼ jQ"j �g! jQj �g:

�

Proof of Lemma 2.9. To ease notation we simply write g", �t and e instead of gi;", �

ti and ei. We only need to check

(C1)–(C3).Step 1. Argument for (C1) and (C3)Consider the periodic function j :¼ atðr�t þ eÞ 2 L2

locðRdÞ, and note that we have �r � j ¼ 0 in D0ðRdÞ by Lemma 2.6

(b). Scaling yields �r � jð �"Þ ¼ 0 in D0ðRdÞ, and thus (C1). Since j is periodic, Proposition 2.10 yields jð �"Þ*>�j ¼

athome weakly in L2locðR

dÞ, and thus (C3).Step 2. Argument for (C2).Since �t and r�t are periodic functions, we conclude from Proposition 2.10 that �tð�"Þ and r�tð�"Þ weakly converge inL2

locðRdÞ, and thus g"ðxÞ ¼ xi þ "�ðx"Þ* xi in H1

locðRdÞ. �

2.2 Stochastic homogenization

Description of Random Coefficients In stochastic homogenization we only have ‘‘uncertain’’ or ‘‘statistical’’information about the coefficient matrix a (which models the microstructure of the material). Hence, faðxÞgx2Rd has tobe considered as a family of matrix valued random variables. For stochastic homogenization the random field a is


required to be stationary in the sense that for any finite number of points x1; . . . ; xk and shift z 2 Rd the random variableðaðx1 þ zÞ; . . . ; aðxk þ zÞÞ has a distribution independent of z, i.e., the coefficients are statistically homogeneous. Inaddition, for homogenization towards a deterministic limit, the random field a is required to be ergodic in the sense thatspatial averages of a over cubes of size R converge to a deterministic constant as R " 1; one could interpret this bysaying that a typical sample of the coefficient field already carries all information about the statistics of the randomcoefficients. For our purpose it is convenient to work within the following mathematical framework:

. We introduce a configuration space of admissible coefficient fields

� :¼ fa : Rd ! Rddsym is measurable and uniformly elliptic in the sense of (2.1)g

. We introduce a probability measure P on � (which we equip with a canonical -algebra). We write h�i for theassociated expectation.

The measure P describes ‘‘the frequency of seeing a certain microstructure in our random material’’. The assumption ofstationarity and ergodicity can be phrased as follows:

Assumption (S). Let ð�;F ;PÞ denote a probability space equipped with the (spatial) ‘‘shift operator’’

: Rd �! �; ðz; aÞ :¼ að� þ zÞ ð¼: zaÞ;

which we assume to be measurable. We assume that the following properties are satisfied:. (Stationarity). For all z 2 Rd and any random variable f 2 L1ð�;PÞ we have

h f � zi ¼ h f i:

. (Ergodicity). For any f 2 L1ð�;PÞ we have

limR"1

?R�

f ðzaÞ dz ¼ h f i for P-a.e. a 2 �: ð2:13Þ

Remark 2.11. Assumption (S) can be rephrased by saying that ð�;F ;P; Þ forms a d-dimensional ergodic, measure-preserving dynamical system. Ergodicity is usually defined as follows: For any E � � (measurable) we have

E is shift-invariant ) PðEÞ 2 f0; 1g:

Here, a (measurable) set E � � is called shift invariant, if zE ¼ E for all z 2 Rd. The fact that this definition ofergodicity implies ð2.13Þ is due to {Birkhoff’s pointwise ergodic theorem}, e.g., see Ackoglu & Krengel [2] forreference that covers the multidimensional case.

Example 2.12 (Random Checkerboard). Let z 2 ð0; 1Þd denote a random vector with uniform distribution, andfakgk2Zd a family of independent, identically distributed random matrices in �0 :¼ fa0 2 Rdd : a0 satisfies (2.1)g.Then

a : Rd ! Rdd; aðxÞ :¼Xk2Zd

1kþzþ�ðxÞak;

(a) independent and identically distributed tiles (b) correlated tiles

Fig. 3. Typical sample of a stationary, ergodic random checkerboard type coefficient field that takes two values with the sameprobability.

12 NEUKAMM

defines random field in � whose distribution is stationary and ergodic. Figures 3(a) shows an example for a sample ofsuch a random field in the case, when a only takes two values, say awhite and ablack. More precisely, the construction ofP is as follows: We start with the probability space ð�0;F 0;P0Þ where F 0 denotes the Borel--algebra on �0 � Rdd,and P0 describes the distribution on a single tile. Then consider the product space

ð�0;F 0;P0Þ :¼ ð�Zd

0 �;F�Zd

0 �Bð�Þ;P�Zd

0 �LÞ;

where L denotes the Lebesgue measure on �, and the map

� : �0 ! �; �ða; zÞ :¼Xk2Zd

1kþzþ�ð�Þak:

The probability measure P is then obtained as the push-forward of P0 under � and yields a stationary and ergodicmeasure. Note that the associated coefficients have a finite range of dependence, in the sense that if we take x; x0 2 Rd

with jx� x0j > diamð�Þ, then the random variables aðxÞ and aðx0Þ are independent, and ð2.13Þ is a consequence of thelaw of large numbers. We might vary the example by considering the convolution

�’ : �Zd

0 ! �Zd

0 ; �’ðaÞk :¼Xj2Zd

’ð j� kÞaj;

with some non-negative convolution kernel ’ : Zd ! R�0 satisfyingP

k2Zd ’ðkÞ ¼ 1. If we define P as the push-forward of P0 under the mapping

�’ : �0 ! �; �ða; zÞ :¼Xk2Zd

1kþzþ�ð�Þ�’ðaÞk;

we obtain again a stationary and ergodic measure. If ’ is not compactly supported, then aðxÞ and aðx0Þ are alwayscorrelated (even for jx� x0j 1), yet they decorrelate on large distances, i.e., for x; x0 2 Zd we have

hðaðxÞ � haiÞðaðx0Þ � haiÞi ¼Xj; j02Zd

’ð j� xÞ’ð j0 � xÞCovðaj; aj0 Þ

� Varða0ÞXj2Zd

’ð j� xÞ’ð j� x0Þ

¼Xj2Zd

’ð jÞ’ð jþ x� x0Þ ! 0 as jx� x0j ! 1:

Figure 3(b) shows a typical sample of a coefficient field obtained in this way (with a kernel ’ that exponentiallydecays).

Example 2.13 (Gaussian random fields). Let f�ðxÞgx2Rd denote a centered, stationary Gaussian random field withcovariance function CðxÞ ¼ Covð�ðxÞ; �ð0ÞÞ. Roughly speaking this means that for any x1; . . . ; xN 2 Rd the randomvector ð�ðx1Þ; . . . ; �ðxNÞÞ has the distribution of a multivariate Gaussian with mean zero and covariance matrix�ij ¼ Cðxi � xjÞ. Suppose that jCðxÞj � ðjxj þ 1Þ�� for some � > 0 (i.e., at least some algebraic decay of correlations).Let � : R! �0 denote a Lipschitz function. Then aðxÞ :¼ �ð�ðxÞÞ defines a stationary and ergodic ensemble ofcoefficient fields.

Problem 8 (Periodic coefficients). Let a# denote a periodic coefficient field in �. Show that there exists a stationaryand ergodic measure P on �, s.t. for any open set O � � we have

Pðfa#ð� þ zÞ : z 2 OgÞ ¼ jOj:

(Hence, with full probability a sample a is a translation of a#. In this sense periodic coefficients can be recast into thestochastic framework).

Homogenization in the stochastic case. The analogue to Theorem 2.2 in the stochastic case is the following:

Theorem 2.14 (Papanicolaou & Varadhan ’79 [27], Kozlov ’79 [20]). Suppose Assumption (S). There exists a(uniformly elliptic) constant coefficient tensor ahom such that for P-a.e. a 2 � we have:For all O � Rd open and bounded, for all f 2 L2ðOÞ, F 2 L2ðO;RdÞ, and " > 0, the unique weak solution u" 2 H1

0ðOÞto

�r � ðaðx"Þru"Þ ¼ f � r � F in O

weakly converges in H1ðOÞ to the weak solution u0 2 H10ðOÞ to

�r � ðahomru0Þ ¼ f � r � F in O;

and we have

að�"Þru" * ahomru0 weakly in L2ðO;RdÞ:


Except for the assumption on a, the statement is similar to Theorem 2.2. Since a 2 � is random, the solutionsu" 2 H1

0ðOÞ (which depend in a nonlinear way on a) are random quantities. In contrast, the homogenized coefficientmatrix ahom is deterministic and only depends on P, but not on the individual sample a, the domain O or the right-handside. Therefore, the limiting equation and thus u0 is deterministic. Hence, in the theorem we pass from an ellipticequation with random, rapidly oscillating coefficients to a deterministic equation with constant coefficients, which is ahuge reduction of complexity. A numerical illustration of the result is given in Fig. 4.

As in the periodic case, the core of the proof is the construction of a corrector (which is then used to define theoscillating test functions in Definition 2.4).

Proposition 2.15 (The corrector in stochastic homogenization). Suppose that Assumption (S) is satisfied. For any� 2 Rd there exists a unique random field � : � Rd ! R, called the corrector associated with �, such that:

(a) For P-a.e. a 2 � the function �ða; �Þ 2 H1locðR

dÞ is a distributional solution to

� r � aðr�ða; �Þ þ �Þ ¼ 0 in D0ðRdÞ; ð2:14Þ

with sublinear growth in the sense that

lim supR!1

1

R2

?R�

j�ða; �Þj2 ¼ 0; ð2:15Þ

and �ða; �Þ is anchored in the sense that>��ða; yÞ dy ¼ 0.

(b) r� is stationary in the sense of Definition 2.21 below.(c) h

>�jr�j2i � 1��2

�2 j�j2, and h>�r�i ¼ 0.

Let us remark that the arguments that we are going to present extend verbatim to the case of systems, see Remark 2.32.Before we discuss the proof of Proposition 2.15, we note that in combination with Lemma 2.9, Proposition 2.15 yieldsa proof of Theorem 2.14. In fact, we only need to show:

Lemma 2.16. Suppose that Assumption (S) is satisfied. For i ¼ 1; . . . ; ei let �i (resp. �ti) denote the correctorassociated with ei and a (resp. the transposed coefficient field at) from Proposition 2.15 and consider the matrixahom 2 Rdd defined by

ahomei :¼?�

aðyÞðr�iða; yÞ þ eiÞ dy� �

:

Then ahom is elliptic and for P-a.e. a 2 � the family ðað �"ÞÞ admits homogenization with homogenized coefficients givenby ahom and oscillating test functions given by

gi;"ðxÞ :¼ xi þ "�tiða;x"Þ:

The proof is a rather direct consequence of the properties of the corrector and ergodicity in form of ð2.13Þ. We present itin Sect. 2.2.1.

(a) (b)

Fig. 4. Illustration of Theorem 2.14 in the case of a random checkerboard-like coefficient field with independent and identicallydistributed tiles. (a) and (b) show realizations of the solutions to �r � ðað �" ru"Þ ¼ 1 in H1

0 ðð0; 1Þ2Þ for " 2 f1

8; 1

32g.

14 NEUKAMM

The main part of this section is devoted to the proof of Proposition 2.15. We first remark that the sublinearitycondition ð2.15Þ is a natural ‘‘boundary condition at infinity’’. Indeed, if the coefficient field a is constant, thensublinearity implies that the solution to ð2.14Þ is unique up to an additive constant.

Lemma 2.17 (A priori estimate for sublinear solutions). Let a 2 �. Suppose u 2 H1locðR

dÞ has sublinear growth inthe sense of ð2.15Þ and solves

�r � aru ¼ 0 in D0ðRdÞ:

Then

lim supR!1

?R�

jruj2 ¼ 0: ð2:16Þ

In particular, if the coefficient field a is constant, then u is constant.

Proof. Let � 2 C1c ðRdÞ. By Leibniz’ rule we have

rðu�Þ � arðu�Þ ¼ rðu�2Þ � aruþ u2r� � ar�þ uðrðu�Þ � ar�� r� � arðu�ÞÞ

Note that by Young’s inequality we have

juðrðu�Þ � ar��r� � arðu�ÞÞj � 2jujjrðu�Þjjr�j ��

2jrðu�Þj2 þ

2

�u2jr�j2;

and thus by ellipticity, and the equation for u,

�

Zjrðu�Þj2 �

Zrðu�Þ � arðu�Þ � 1þ

2

�

� �Zu2jr�j2 þ

�

2

Zjrðu�Þj2:

We conclude that Zjrðu�Þj2 � CðdÞ

Zu2jr�j2:

We now specify the cut-off function: Let �1 2 C1c ð2�Þ satisfy �1 ¼ 1 on �. Then the above estimate applied with�1ð �RÞ yields ?

R�

jruj2 � R�dZjrðu�1ð �RÞÞj

2 � Cðd; �ÞR�d�2

Zu2jðr�1Þð �RÞj

2

� Cðd; �; �1Þð2RÞ�2

?2R�

juj2:

By sublinearity, for R!1 the right-hand side converges to 0, which yields the first claim. If a is constant, then by astandard interior estimate we have

kruk2L1ðR�Þ � C

?2R�

jruj2;

for a constant C that is independent of R. We conclude that ru ¼ 0 a.e. in Rd and thus u is constant. �

Let us anticipate that the above estimate also yields uniqueness in the case of stationary and ergodic randomcoefficients for solutions with sublinear growth and stationary gradients, see Corollary 2.26 below. On the other hand itis not clear at all that the equation �r � ðar�Þ ¼ r � ða�Þ admits a sublinear solution. In fact, this is only true for‘‘generic’’ coefficient fields a 2 �, in particular, we shall see that this is true for P-a.e. a 2 � when P is stationary andergodic. Our strategy is the following:

. Instead of the equation �r � ðar�Þ ¼ r � ða�Þ we consider the modified corrector equation

1

T�T �r � aðr�T Þ ¼ r � ða�Þ in Rd ðT 1Þ; ð2:17Þ

which turns out to be well-posed for all a 2 � and yields an a priori estimate of the form

8R � T :

?ffiffiffiRp�

1

Tj�T j2 þ jr�T j2 � Cðd; �Þj�j2: ð2:18Þ

. By stationarity of P we can turn ð2.18Þ into an averaged estimate that on the level of r�T is uniform in T ,?�

jr�T j2� �

� Cðd; �Þj�j2:

. This allows us to pass to the weak limit (for T " 1) in an appropriate subspace of random fields. The limit � is asolution to the corrector equation, its gradient is stationary, i.e., r�ða; xþ zÞ ¼ r�ðxa; yÞ and satisfies


?�

jr�j2� �

<1 and

?�

r��

¼ 0:

. Finally, by exploiting ergodicity and the property that h>�r�Ti ¼ 0 we deduce sublinearity.

We start with the argument that establishes sublinearity, since the latter is the most interesting property of the corrector.In fact, the argument can be split into a purely deterministic argument (that we state next), and a non-deterministic partthat exploits ergodicity, see proof of Corollary 2.27 below.

Lemma 2.18 (sublinearity). Let u 2 H1locðR

dÞ satisfy>�u ¼ 0,

lim supR!1

?R�

jruj2 <1; and ð2:19Þ

lim supR!1

?R�

ruðyÞ � FðRxÞ ¼ 0 for all F 2 L2ð�;RdÞ: ð2:20Þ

Then we have

lim supR!1

1

R2

?R�

u�?

R�

u

��2

!12

¼ 0; ð2:21Þ

lim supR!1

1

R2

?R�

juj2� �1

2

¼ 0: ð2:22Þ

Proof. Step 1. Proof of ð2.21Þ.We appeal to a scaling argument: Consider uRðxÞ :¼ 1

RðuðRxÞ �

>R�uÞ and note that

ruRðxÞ ¼ ruðRxÞ;Z�

juRj2 ¼1

R2

?R�

u�?

R�

u

��2:

Hence, it suffices to show that uR ! 0 strongly in L2ð�Þ. Since>�uR ¼ 0, Poincare’s inequality yieldsZ

�

u2R �

Z�

jruRj2 ¼?

R�

jruj2;

and ð2.19Þ implies that ðuRÞ is bounded in H1ð�Þ. By weak compactness of bounded sequences in H1ð�Þ, we findu1 2 H1ð�Þ such that uR * u1 weakly in H1ð�Þ (for a subsequence that we do not relabel). Since H1ð�Þ � L2ð�Þ iscompactly embedded (by the theorem of Rellich–Kondrachov), we may assume w.l.o.g. that we also have uR! u1strongly in L2ð�Þ. We claim that u1 ¼ 0 (which then also implies that the convergence holds for the entire sequence).Indeed, from ð2.20Þ we deduce that?

�

ru1 � F ¼ limR!1

?�

ruR � F ¼ limR!1

?�

ruðRxÞ � FðxÞ ¼ 0:

Hence, ru1 ¼ 0, and thus u1 is constant. Since>�u1 ¼ 0, u1 ¼ 0 follows.

Step 2. Proof of ð2.22Þ.Set JðtÞ ¼

>t�u ¼

>�uðtxÞdx. We have

@tJðtÞ ¼?�

ruðtxÞ � xdx:

Let R T � 1. Then?R�

u�?�

u

�� ¼

Z R

1

@tJðtÞdt��

�� Z T

1

j@tJðtÞjdt þZ R

T

j@tJðtÞjdt

¼Z T

1

?�

ruðtxÞ � x��

��dt þZ R

T

?�


��dt� CðdÞ

Z T

1

?�

jruðtxÞj2dx� �1

2

þ ðR� TÞ supt�T

?�


��:By ð2.19Þ Z T

1

?�

jruðtxÞj2dx� �1

2

¼Z T

1

?t�

jruðxÞj2dx� �1

2

<1:

Hence, dividing by R and taking the limit R!1 yields

16 NEUKAMM

lim supR!1

1

R

?R�

u�?�

u

�� sup

t�T

?�


��:By ð2.20Þ (applied with FðxÞ ¼ x), in the limit T !1, the last expression converges to 0. We conclude

1

R2

?R�

juj2� �1

2

�1

R2

?R�

u�?

R�

u

��2

!12

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}!0

þ1

R

?R�

u�?�

u

��|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}

!0

þ1

R

?�

u

��|fflfflfflffl{zfflfflfflffl}

¼0

:

�

Remark 2.19. Let us anticipate that in the proof of Proposition 2.15 we apply Lemma 2.18 in the special situationwhere u is a realization of a random field u : � Rd ! R, whose gradient is stationary and satisfies h

>�jruj2i <1

and h>�rui ¼ 0. Then, properties ð2.19Þ and ð2.20Þ hold for uða; �Þ for P-a.e. a 2 � as we will prove by appealing to

ergodicity ð2.13Þ.

Another argument that is purely deterministic is the existence theory for the modified corrector. Note that the right-hand side of ð2.17Þ is a divergence of a vector field F : Rd ! Rd that is not integrable (yet bounded). For thedeterministic a priori estimate it is convenient to consider the weighted norm

kFk2� :¼ZjFðxÞj2�ðxÞ dx; ð2:23Þ

where � : Rd ! R denotes a positive, exponentially decaying weight to be specified below. In the following variousestimates are localized on cubes. We use the notation

Q :¼ fQ ¼ xþ r� : x 2 Rd; r > 0g; � :¼ �1

2;1

2

� �d

:

Lemma 2.20. There exists a positive, exponentially decaying weight � withRRd � ¼ 1 (that only depends on d and �)

and a constant C ¼ Cðd; �Þ, such that the following properties hold: Let a 2 �, T > 0, F 2 L2locðR

d;RdÞ with kFk� <1 (cf. ð2.23Þ). Then there exists a unique solution u 2 H1

locðRdÞ to

1

Tu� r � aru ¼ r � F in D0ðRdÞ; ð2:24Þ

such that R 7!>R�juj2 grows at most polynomially for R!1. The solution satisfies the a priori estimate

8R � T :

?ffiffiffiRp�

1

Tjuj2 þ jruj2 � Cðd; �ÞkFð

ffiffiffiRp�Þk2� : ð2:25Þ

Proof. Step 1. Proof of the a priori estimate.We claim that u satisfies ð2.25Þ. For the argument let � 2 C1c ðR

dÞ. By Leibniz’ rule we have

rðu�Þ � arðu�Þ ¼ rðu�2Þ � aruþ u2r� � ar�þ uðrðu�Þ � ar�� r� � arðu�ÞÞ:

Note that

juðrðu�Þ � ar��r� � arðu�ÞÞj � 2jujjrðu�Þjjr�j:

Thus, integration, ellipticity, and ð2.24Þ yieldZ1

Tju�j2 þ �jrðu�Þj2 �

Z1

Tju�j2 þ rðu�Þ � arðu�Þ

�Z

1

Tðu�2Þuþrð�2uÞ � aruþ

Zu2jr�j2

þ 2

Zjrðu�Þjjujjr�j

¼Z

F � rðu�2Þ þZ

u2jr�j2 þ 2

Zjrðu�Þjjujjr�j

�ZjFjj�jjrðu�Þj þ jFjjujj�jjr�j þ

Zu2jr�j2

þ 2

Zjrðu�Þjjujjr�j:

With Young’s inequality in form of


jFjjujj�jjr�j �1

2jFj2�2 þ

1

2u2jr�j2;

jFjj�jjrðu�Þj �1

�jFj2�2 þ

�

4jrð�uÞj2;

2jrðu�Þjjujjr�j �4

�u2jr�j2 þ

�

4jrð�uÞj2;

we get

1

T

Zju�j2 þ

�

2jrðu�Þj2 � c�

ZjFj2�2 þ

Zu2jr�j2

� �; c� :¼

4

�þ

3

2

� �:

Let R � T . By an approximation argument (that exploits that R 7!>R�juj2 grows at most polynomially), this estimate

extends to the exponential cut-off function �ðxÞ ¼ expð�c0jxjffiffiffiRp Þ, where c0 :¼ 1

2ffiffiffiffiffidc�p to the effect of

c� jr�j2

�2�

dc�c20

R�

1

2R�

1

2T:

We conclude that Z1

2Tju�j2 þ

�

2jrðu�Þj2 � c�

ZjFj2�2;

and thus Z1

Tjuj2 þ jruj2

� ��2 � Cðd; �Þ

ZjFj2�2: ð2:26Þ

Since min ffiffiffiRp��2 � expð�2c0Þ > 0, we deduce that?

ffiffiffiRp�

1

Tjuj2 þ jruj2

� �� Cðd; �ÞR�

d2

ZjFj2�2:

On the other hand, with

�ðxÞ :¼Z

expð�2c0jyjÞ dy� ��1

expð�2c0jxjÞ;

we may estimate the right-hand side of the previous estimate byZjFj2�2 � Cðd; �ÞR

d2

ZjFð

ffiffiffiRp�Þj2�ð�Þ;

and thus obtain ð2.25Þ.Step 2. Conclusion.Consider a general right-hand side F with kFk� <1. For k 2 N set FkðxÞ :¼ 1ðjxj < kÞFðxÞ, which is a vector field inL2ðRd;RdÞ. Therefore, by the theorem of Lax–Milgram we find uk 2 H1ðRdÞ that solves

1

Tuk �r � aruk ¼ r � Fk;

and satisfies the standard a priori estimate,

1

T

Zu2k þ

�

2jrukj2 �

2

�

ZjFkj2: ð2:27Þ

In particular, R 7!>R�jukj2 is bounded and thus uk satisfies the a priori estimate of Step 1,

8R � T :

?ffiffiffiRp�

1

Tjukj2 þ jrukj2

� �� Cðd; �ÞkFkk2� � Cðd; �ÞkFk2� ;

which is uniform in k. Consider the nested sequence of cubes Q‘ :¼ 2‘ffiffiffiffiTp�, ‘ 2 N0. By the a priori estimate we

conclude that ðukÞ is bounded in H1ðQ‘Þ for any ‘ 2 N0. Since Q‘ � Q‘þ1 and Q‘ " Rd, we conclude that there existsu 2 H1

locðRdÞ such that uk * u weakly in H1ðQ‘Þ for any ‘ 2 N0. Consequently u solves ð2.24Þ in a distributional sense.

Thanks to the lower-semicontinuity of the norm, we deduce that u satisfies the a priori estimate ð2.25Þ. This proves theexistence of the solution. Uniqueness of u is a consequence of the a priori estimate. �

As already mentioned, in order to obtain an estimate that is uniform in T , we need to exploit stationarity of P andrandom fields in the following sense:

Definition 2.21 (Stationary random field). A measurable function u : � Rd ! R is called a stationary L1-randomfield (or short: stationary), if h

>Qjuji <1 for all cubes Q � Q and if for P-a.e. a 2 �,

18 NEUKAMM

ZxþQ

uða; yÞ dy ¼ZQ

uðxa; yÞ dy for all cubes Q 2 Q and x 2 Rd: ð2:28Þ

A prototypical example of a stationary random variable is as follows: Take u0 2 L1ð�Þ and consider uða; xÞ :¼ u0ðxaÞ.Then u is a stationary L1-random field, called the stationary extension of u0. One can easily check that for any A � Rd

open and bounded we have ?A

uða; yÞ dy� �

¼?

A

hu0ðyaÞi ¼ hu0i;

where the last identity holds by stationarity of P. In particular, we deduce that the value of h>Quða; yÞ dyi, Q 2 Q, is

independent of Q. The same properties are true for general stationary L1-random fields (except for the difference thatwe need to invoke an average w.r.t. Rd-component to obtain well-defined quantities):

Lemma 2.22. Suppose P is stationary. Let f denote a stationary L1-random field.(a) For any A � Rd open and bounded we have ?

A

f

� �¼

?�

f

� �:

(b) Let � > 0 and set f�ðaÞ :¼>�� f ða; yÞ dy. Then f� 2 L1ð�Þ and for P-a.e. a 2 �,?

��f ða; xþ yÞ dy ¼ f�ðxaÞ for all x 2 Rd:

We postpone the proof to Sect. 2.2.1. As a consequence of ð2.13Þ (i.e., Birkhoff’s ergodic theorem), we obtain thefollowing variant for stationary fields:

Lemma 2.23 (Variant of Birkhoff’s ergodic theorem). Let f denote a stationary L1-random field, then for P-a.e.a 2 � we have

limR!1

?R�

f ða; xÞ dx ¼?�

f

� �:

Moreover, if additionally h>�jf j2i <1, then

limR!1

?�

f ða;RxÞ�ðxÞ dx ¼?�

f

� �?�

� dx for all � 2 L2ð�Þ:

We postpone the proof to Sect. 2.2.1.We turn back to the modified corrector equation which corresponds to the equation ð2.24Þ with right-hand side

Fða; xÞ :¼ aðxÞ�. This random field (and by uniqueness the associated solution) is stationary. Hence, as a corollary ofLemma 2.20 and Lemma 2.22 we obtain:

Corollary 2.24. Suppose P is stationary. Let T � 1 and � 2 Rd. Then there exists a unique stationary random field�T that solves the modified corrector equation

1

T�T � r � ðaðr�T þ �ÞÞ ¼ 0 in D0ðRdÞ, P-a.s.; ð2:29Þ

and which satisfies the a priori estimate ?�

1

T�2T þ jr�T j

2

� �� Cðd; �Þj�j2:

Moreover, we have ?�

r�T� �

¼ 0:

Proof. Step 1. Existence and a priori estimate.For a 2 � let �T ða; �Þ denote the solution to ð2.24Þ with F ¼ Fða; �Þ ¼ að�Þ� of Lemma 2.20. By uniqueness of thesolution we deduce that �T is stationary. Hence, by Lemma 2.22,?

�

1

T�2T þ jr�T j

2

� ��

?ffiffiffiTp�

1

T�2T þ jr�T j

2

� �� Cðd; �ÞhkFð

ffiffiffiffiTp�Þk2�i;

and the estimate follows, since


hkFðffiffiffiffiTp�Þk2�i ¼

Zhjað

ffiffiffiffiTp

xÞ�j2i�ðxÞ dx � j�j2:

Step 2. Zero expectation of the gradient.This is in fact a general property of stationary random fields u satisfying h

>�u2 þ jruj2i <1. Indeed, by stationarity

and the divergence theorem we have

I :¼?�

@iuða; xÞ dx� �

¼?�

?�

@iuða; xþ yÞ dx� �

dy

¼1

j�j

?�

Z@�

uða; xþ yÞ iðxÞ dSðxÞ� �

dy;

where ðxÞ denotes the outer unit normal at x 2 @�. By Fubini we may switch the order of the integration and get

I ¼1

j�j

Z@�

?�

uða; xþ yÞ dy� �

iðxÞ dSðxÞ ¼1

j�j

?�

uða; yÞ dy� � Z

@�

iðxÞ dSðxÞ

¼ 0;

where in the second last step we used stationarity, and in the last stepR@� iðxÞ dSðxÞ ¼ 0.

�

The estimate on r�T of Corollary 2.24 is uniform T . Motivated by this we introduce a suitable function space inwhich we can pass to the limit T !1. Since we can only pass to the limit on the level of the gradient, it is convenientto consider uT ¼ �T �

>��T , which satisfies

>�uT ¼ 0, and thus is uniquely determined by ruT ¼ r�T .

Lemma 2.25. Suppose P is stationary. Consider the linear space

H :¼ u : � Rd ! R :

?�

juj2 þ jruj2� �

<1;?�

u

��

� �¼ 0;ru is stationary

:

Then,(a) for any cube Q 2 Q we have ?

Q


� Cðd;QÞ?�

jruj2� �

:

(b) H equipped with the inner product

ðu; vÞH :¼?�

ru � rv� �

is a Hilbert space.

Proof. Step 1. Proof of (a).We start with a deterministic estimate. Consider the dyadic family of cubes Qn ¼ 2n�, n ¼ 0; 1; . . . . We claim that

?Qn

juj2� �1

2

� CðdÞXn‘¼1

2‘?

Q‘

jruj2� �1

2

: ð2:30Þ

Indeed,

?Qn

juj2� �1

2

�?

Qn

u�?

Qn�1

u

��2

!12

þ?

Qn�1

u

��

�?

Qn

u�?

Qn�1

u

��2

!12

þ?

Qn�1

juj2� �1

2

�Xn‘¼1

?Q‘

u�?

Q‘�1

u

��2

!12

þ?�

u

��|fflffl{zfflffl}

¼0

� CðdÞXn‘¼1

2‘?

Q‘

jruj2� �1

2

:

Suppose u 2 H. Taking the square and expectation of ð2.30Þ, and exploiting stationarity in form of

20 NEUKAMM

?Q‘

jruj2� �

¼?�

jruj2� �

;

yields ?Qn

juj2� �

� CðdÞnXn‘¼1

22‘

?Q‘

jruj2� �

� Cðd; nÞ?�

jruj2� �

: ð2:31Þ

Now, let Q denote an arbitrary cube. Then we have Q � Qn for some n 2 N, and thus?Q


¼?

Q

juj2� �

þ?�

jruj2� �

� C0ðd; nÞ?�

jruj2� �

:

Step 2. H is Hilbert.Obviously ð�; �ÞH turns H into an inner product space and the definiteness of the norm follows from (a). We argue thatðH; k � kHÞ is complete. First note that by stationarity of ru, we have for all n 2 N,?

Qn

jruj2� �

¼?�

jruj2� �

¼ kuk2H ; ð2:32Þ

and thus by Step 1, ?Qn


� Cðd; nÞkuk2H:

Let ðukÞ denote a Cauchy sequence in H. Then the previous estimate implies that ðukÞ is Cauchy in any of the spacesL2ð�;H1ðQnÞÞ, n 2 N. Thus, uk ! uðnÞ in L2ð�;H1ðQnÞÞ for all n 2 N0. For ‘ � n, we have Q‘ � Qn, and thusuð‘Þ ¼ uðnÞ on � Q‘. We conclude that there exists a random field u with u 2 L2ð�;H1ðQÞÞ for all cubes Q 2 Q, anduk ! u in L2ð�;H1ðQÞÞ for any Q 2 Q. This in particular implies that hj

>�uji ¼ 0. To conclude u 2 H it remains to

argue that ru is stationary. It suffices to show for any ’ 2 L2ð�Þ, Q 2 Q and x 2 Rd,?xþQ

@iuða; yÞ dy’ðaÞ� �

¼?

Q

@iuðxa; yÞ dy’ðaÞ� �

:

Since @iuk is stationary, this identity is satisfied for u replaced by uk. Since @iuk ! @iu in L2ð� QÞ for any Q 2 Q, theidentity also holds for @iu. �

As a corollary of Lemma 2.25, Corollary 2.24 and Lemma 2.18 we obtain the existence and uniqueness of thesublinear corrector:

Corollary 2.26 (Uniqueness of the sublinear corrector). Suppose P is stationary and ergodic. Then there exists atmost one � 2 H satisfying the corrector equation ð2.14Þ and the sublinear growth condition ð2.15Þ P-a.s.

Proof. Let �; �0 2 H be two sublinear solutions to the corrector equation and consider u :¼ �� 0. Then P-a.s. usatisfies the assumptions of Lemma 2.17 and we conclude

limR!1

?R�

jruj2 ¼ 0:

On the other hand, by stationarity of ru and ergodicity we have?�

jruj2� �

¼ limR!1

?R�

jruj2 ¼ 0;

and thus u is constant P-a.s. Since>�u ¼ 0, we conclude that u ¼ 0. �

Corollary 2.27 (Existence of the sublinear corrector). Suppose P is stationary and ergodic. Let �T denote thesolution to the modified corrector equation ð2.29Þ of Corollary 2.24. Then there exists � 2 H such that uT :¼�T �

>��T * � weakly in H (for T !1), and � is the unique solution to the corrector equation in the sense of

Corollary 2.26.

Proof. By Corollary 2.24 ðuT Þ is a bounded sequence in H. Since H is Hilbert, we may pass to a subsequence (notrelabeled) such that uT * � weakly in H. We claim that � solves the corrector equation ð2.14Þ. Let � 2 C1c ðR

dÞ and’ 2 L2ð�Þ denote test functions and let Q 2 Q denote a cube centered at 0 with supp � � Q. From uT * � weakly inH, we infer that r�T * r� weakly in L2ð� QÞ, and thus

’

Zaðr�þ �Þ � r�

� �¼ lim

T!1’

Zaðr�T þ �Þ � r�

� �¼ � lim

T!1’

Z1

T�T�

� �:

Note that


Z1

T�T�

�� jQj 1

T

?Q

�2T

� �12 1

T

?Q

�2

� �12

: ð2:33Þ

By stationarity and the a priori estimate of Corollary 2.24 we have

’

Z1

T�T�

� �� T�

12 jQj

?Q

�2

� �12

h’2i12

1

T

?�

�2T

� �12

! 0;

and thus we deduce with ð2.33Þ that

’

Zaðr�þ �Þ � r�

� �¼ 0:

Since the test functions are arbitrary, ð2.14Þ follows. Since � 2 H, we have h>�jr�j2i <1, h

>�r�i ¼ 0, and

hj>��ji ¼ 0. By ergodicity, which we use in form of Lemma 2.23, we find that the assumptions of Lemma 2.18 are

satisfied P-a.s. Hence, �ða; �Þ is sublinear in the sense of ð2.15Þ P-a.s., and a solution to ð2.14Þ. By uniqueness of thesolution (cf. Corollary 2.26) we conclude that � is independent of the subsequence, and we deduce that uT * � in H

for the entire sequence. �

Note that Corollary 2.27 proves Proposition 2.15 except for the a priori estimate?�

jr�j2� �

�1� �2

�2j�j2; ð2:34Þ

whose argument we postpone to the end of this section. In fact, the estimate h>�jr�j2i � Cðd; �Þj�j2 (for some constant

Cðd; �Þ <1) follows (by lower semicontinuity) directly from the a priori estimate in Corollary 2.24. The sublinearcorrector of Proposition 2.15 can alternatively be characterized as the unique solution to an abstract variationalproblem in the Hilbert space H. (This formulation also entails a short argument for ð2.34Þ). In the rest of this section,we discuss this alternative formulation. We start with the observation that the space of stationary H1-random fieldsforms a Hilbert space:

Lemma 2.28. Suppose P is stationary. Consider the linear space

S :¼ u is a stationary random field with

?�


<1

:

Then S with inner product

ðu; vÞS :¼?�

uvþ ru � rv� �

is a Hilbert space. Moreover, for any u 2 S we have h>�rui ¼ 0.

Proof. Obviously ð�; �ÞS turns S into an inner product space. We argue that ðS; k � kSÞ is complete and first note that forany u 2 S, the stationarity of u implies stationarity of ru, and thus for all Qn :¼ 2n�, n 2 N, we have?

Qn

u2 þ jruj2� �

¼?�

u2 þ jruj2� �

¼ kuk2S: ð2:35Þ

The remaining argument for completeness is similar to the proof of Lemma 2.25. The fact that gradients of stationaryrandom fields are mean-free has already been proven in Step 2 in the proof of Corollary 2.24. �

Next we observe that on the level of the gradient any function u 2 H can be approximated by functions in S. Withhelp of this observation we can pass from distributional equations on Rd to problems in H (and vice versa):

Lemma 2.29. Suppose P is stationary and ergodic.(a) For any u 2 H we can find a sequence uT 2 S such that uT �

>�uT * u weakly in H.

(b) Let F be a stationary random vector field with h>�jFj2i <1. Then the following are equivalent?

�

F � r’� �

¼ 0 for all ’ 2 H; ð2:36Þ

�r � F ¼ 0 in D0ðRdÞ, P-a.s. ð2:37Þ

Proof of Lemma 2.29. Step 1.Let F denote a stationary vector field with h

>�jFj2i <1, let T � 1. We claim that there exists a unique uT 2 S such

that

1

TuT �4uT ¼ r � F in D0ðRdÞ, P-a.s.; ð2:38Þ

22 NEUKAMM

and that uT is characterized by the weak equation?�

1

TuT’þ ruT � r’

� �¼ �

?�

F � r’� �

for all ’ 2 S: ð2:39Þ

We first argue that a solution uT 2 S to ð2.38Þ exists. Note that by stationarity we have for all R � 1,

hkFðffiffiffiRp�Þk2�i ¼

?�

jFj2� �

:

Thus, by Lemma 2.20, there exists a unique random field uT that satisfies ð2.38Þ and the a priori bound ð2.25Þ P-a.s.Since F is stationary, uT and ruT are stationary, and thus the a priori bound turns into?

�

1

Tu2T þ jruT j

2

� �� Cðd; �Þ

?�

jFj2� �

: ð2:40Þ

On the other hand, the Lax–Milgram Theorem yields a unique solution vT 2 S to the weak formulation ð2.39Þ. In orderto conclude that both formulations are equivalent, it suffices to show that uT solves ð2.39Þ. For the argument let ’ 2 S

and � 2 C1c ð�Þ be arbitrary test functions. It suffices to show

I :¼?�

1

TuT’þ ðruT þ FÞ � r’

� �¼ 0:

For R � 1 set ’R :¼ 1R’ðR�Þ, uT ;R :¼ 1

RuT ;RðR�Þ, and FR :¼ FðR�Þ. Then by stationarity and scaling we have

I ¼?�

R2

TuT ;R’R þ ðruT ;R þ FRÞ � r’R

� �;

and by ð2.38Þ, ?�

R2

TuT ;Rð’R�Þ þ ðruT ;R þ FRÞ � rð’R�Þ

� �¼ 0:

The difference of the previous two equations is given by?�

R2

TuT ;R’Rð1� �Þ

� �þ ðruT ;R þ FRÞ � ðr’R �rð’R�Þ� �

¼: II þ III:

By Cauchy–Schwarz, stationarity, and the a priori estimate ð2.40Þ,

jIIj �1ffiffiffiffiTp

?�

1

TjuT ðR�Þj2

� �12

?�

j’ðR�Þj2j1� �j2� �1

2

� CðdÞ1ffiffiffiffiTp

?�

jFj2� �1

2?�

’2

� �12Z�

j1� �j2� �1

2

� Cðd;T ;F; ’Þk1� �kL2ð�Þ:

Regarding III we note that

jIIIj �?�

jruR þ FRjðjr’Rjj1� �j þ j’Rjjjr�jÞ� �

Arguing as above, we deduce that

jIIIj � Cðd;F; ’Þðk1� �kL2ð�Þ þ kr�kL1ð�Þhk’Rk2L2ð�Þi12Þ:

Note that by stationarity we have

hk’Rk2L2ð�Þi ¼ R�2

?R�

j’j2� �

¼ R�2

?�

j’j2� �

! 0:

In conclusion we deduce that

jIj � lim supR!1

ðjIIj þ jIIIjÞ � Cðd;T ;F; ’Þk1� �kL2ð�Þ:

Since � is arbitrary, the right-hand side can be made arbitrarily small, and thus I ¼ 0.Step 2. Proof of (a).Let u 2 H, set Fða; xÞ :¼ �ruða; xÞ, and let uT denote the unique solution in S to ð2.38Þ. From ð2.39Þ we obtain the apriori estimate


?�

1

TjuT j2 þ

1

2jruT j2

� ��

1

2

?�

jruj2� �

; ð2:41Þ

which for the gradient is uniform in T � 1. We conclude that vT :¼ uT �>�uT defines a bounded sequence in H. Let

v 2 H denote a weak limit of ðvT Þ along a subsequence T !1 (that we do not relabel). We claim that v ¼ u (whichimplies that the convergence holds for the entire sequence). First notice that it suffices to show that for all ’ 2 L2ð�Þand � 2 C1c ðR

dÞ we have

’

Zðrv� ruÞ � r�

� �¼ 0: ð2:42Þ

Indeed, this implies that w ¼ v� u satisfies �4w ¼ 0 in D0ðRdÞ, P-a.s. Since w 2 H has sublinear growth, weconclude with Lemma 2.17 that w is constant. Since

>�w ¼ 0 by construction, we deduce that w ¼ v� u ¼ 0. We

prove ð2.42Þ. Since rv is a weak limit of rvT ¼ ruT , it suffices to show that

I :¼ ’

?Q

ðruT � ruÞ � r��

! 0 for T !1;

where Q 2 Q is a cube that contains the support of �. Since uT solves ð2.38Þ with F ¼ �ru, we have

I ¼ � ’

?Q

1

TuT�

� �;

which for T !1 converges to 0, thanks to the a priori estimate ð2.41Þ and stationarity.Step 4. Proof (b).First note that ð2.36Þ, thanks to (a), is equivalent to?

�

F � r’� �

¼ 0 for all ’ 2 S: ð2:43Þ

Let uT 2 S denote the unique solution to

1

TuT �4uT ¼ �r � F in D0ðRdÞ, P-a.s.;

which exists thanks to Step 2, and is equivalent to?�

1

TuT’þ ðruT � FÞ � r’

� �¼ 0 for all ’ 2 S: ð2:44Þ

Then for all T � 1, ð2.37Þ is equivalent to uT ¼ 0. On the other hand, in view of ð2.44Þ, uT ¼ 0 implies ð2.43Þ, andð2.43Þ implies ?

�

1

TuT’

� �¼ 0 for all ’ 2 S;

and thus uT ¼ 0. �

Finally, we present the characterization of � and �T by means of variational problems in the Hilbert spaces H and S:

Lemma 2.30. Suppose P is stationary and ergodic. Let � denote the sublinear corrector associated with � 2 Rd ofProposition 2.15, and �T the unique modified corrector associated with � 2 Rd of Corollary 2.24. Then � 2 H and�T 2 S are uniquely characterized by ?

�

aðr�þ �Þ � r’� �

¼ 0 for all ’ 2 H; ð2:45Þ?�

1

T�T’þ aðr�T þ �Þ � r’

� �¼ 0 for all ’ 2 S; ð2:46Þ

and we have

limT!1

?�

jr�T � r�j2� �

¼ 0:

Moreover, ð2.34Þ holds.

Proof. First note that the variational equations for � and �T in H and S, respectively, admit a unique solution by thetheorem of Lax–Milgram. The equivalence of the formulations for � follows from Lemma 2.29 (b). The equivalence ofthe formulation for �T follows by the argument in Step 1 in the proof of Lemma 2.29. (We only need to replace �4 by�r � ðarÞ and F by a�). For the convergence statement it is convenient to work with the variational equations:

24 NEUKAMM

?�

ðr��r�T Þ � aðr��r�T Þ� �

¼?�

ðr�� r�T Þ � aðr�þ �Þ� �

�?�

ðr� � aðr�T þ �Þ� �

þ?�

ðr�T � aðr�T þ �Þ� �

¼ �?�


�1

T

?�

�2T

� �:

Hence,

lim supT!1

?�

ðr�� r�T Þ � aðr�� r�T Þ� �

� � limT!1

?�


¼ �?�

r� � að�þr�Þ� �

¼ 0;

and the claim follows by ellipticity of a. The a priori estimate for r� easily follows from the variational formulation ofthe corrector equation: We first note that by ellipticity and ð2.45Þ, we have

�

?�

jr�þ �j2� �

�?�

ðr�þ �Þ � aðr�T þ �Þ� �

¼ � �?�

aðr�T þ �Þ� �

�1

2�j�j2 þ

�

2

?�

jr�T þ �j2� �

;

and thus

�

2

?�

jr�þ �j2� �

�1

2�j�j2:

On the other hand, ?�

jr�þ �j2� �

¼?�

jr�j2� �

þ j�j2;

since the cross-term h>�r� � �i ¼ 0, thanks to h

>�r�i ¼ 0. Thus, ð2.34Þ follows from the combination of these

estimates. �

Note that Corollary 2.27 combined with ð2.34Þ, which follows from the previous lemma, completes the proof ofProposition 2.15. As a corollary of the previous lemma, and in analogy to Lemma 2.8, we have:

Lemma 2.31 (Properties of the homogenized coefficients). Suppose Assumption (S) is satisfied and let �1; . . . ; �ddenote the correctors associated with e1; . . . ; ed. Set

ahomei :¼?�

aðr�i þ eiÞ� �

:

Then:(a) (ellipticity). For any � 2 Rd we have

� � ahom� � �j�j2:

(b) (invariance under transposition). Let �ti denote the corrector associated with the transposed matrix at. Then

ðahomÞtei ¼?�

atðr�ti þ eiÞ� �

:

(c) (symmetry). If a is symmetric (a.e. in Rd and P-a.s.), then ahom is symmetric.

The proof is similar to the proof of Lemma 2.8. We leave it to the reader.

Remark 2.32 (Systems). The arguments that we presented in this section (in particular the construction of thesublinear corrector and the proof of Theorem 2.14 extend to systems of the form

�r � aru ¼ F;

with u : Rd ! H taking values in a finite dimensional Euclidean space H. The matrix field a : Rd ! LinðHd ;HdÞ isrequired to be bounded and uniformly elliptic in the integrated form ofZ

r� � ar� � �ZjD�j2; for all � 2 C1c ðR

d;HÞ:

In particular, this includes the relevant case of linear elasticity, when H ¼ Rd and a : Rd ! LinðRdd;RddÞ is Korn-


elliptic, i.e.,

� � aðxÞ� � jsym �j2 for a.e. x 2 Rd and � 2 Rdd:

2.2.1 Proof of Lemma 2.16, Lemma 2.22, and Lemma 2.23

Proof of Lemma 2.16. Set ji :¼ atðr�ti þ eiÞ and note that

jiðx"Þ ¼ at"rgi;":

Note that by the corrector equation, we have �r � ji ¼ 0, and thus property (C1) holds. Since ji is a stationary randomfield with h

>�jjij2i <1, Birkhoff’s ergodic theorem in form of Lemma 2.23 implies that jið�"Þ* h

>�jii ¼ ahomei, and

thus property (C3) is satisfied. Finally, since �ti has sublinear growth, we deduce that?Q

j"�tið�"Þj

2! 0 for all Q 2 Q and P-a.s.;

and thus property (C2) is satisfied. �

Proof of Lemma 2.22. Step 1. Proof of (a).We first claim that for any Q 2 Q centered at 0, and any odd ‘ 2 N we have?

‘Qf

� �¼

?Q

f

� �: ð2:47Þ

Indeed, since with s denoting the side length of Q, we have ‘Q ¼ [x2sZd\‘Qðxþ QÞ, up to a set of zero measure, we getby stationarity of f and P:?

‘Qf

� �¼

Xx2sZd\‘Q

jxþ Qjj‘Qj

?xþQ

f

� �¼ ‘�d

Xx2sZd\‘Q

?Q

f ðxa; yÞ dy� �

¼ ‘�dX

x2sZd\‘Q

?Q

f ða; yÞ dy� �

¼?

Q

f

� �:

Next we prove (a) for any Q 2 Q with Q centered at 0. W.l.o.g. we may assume that f � 0. (Otherwise decompose f inits positive and negative part, which remain stationary). For ‘ 2 N0 let ‘� (and ‘þ) denote the largest (smallest) oddnon-negative integer satisfying

‘�� ‘Q � ‘þ�;

and note that

j‘��jj‘Qj

! 1 as ‘!1:

Thus Z‘��

f �Z‘Q

f �Z‘þ�

f ;

dividing by j‘Qj and taking the expectation yields

j‘��jj‘Qj

?‘��

f

� ��

?‘Q

f

� ��j‘þ�jj‘Qj

Z‘þ�

f

� �;

By ð2.47Þ we have ?‘��

f

� �¼

?‘þ�

f

� �¼

?�

f

� �;

and ?Q

f

� �¼

?‘Q

f

� �;

and thus

j‘��jj‘Qj

?�

f

� ��

?Q

f

� ��j‘þ�jj‘Qj

Z�

f

� �:

Taking the limit ‘!1 yields (a) for centered cubes. The conclusion for an arbitrary cube Q 2 Q follows directlyfrom the stationarity of f and P: If we choose x 2 Rd such that xþ Q is centered, then

26 NEUKAMM

?Q

f

� �¼

?xþQ

f ð�xa; yÞ dy� �

¼?�

f

� �:

The statement for an arbitrary open, bounded set A � Rd follows from Whitney’s covering theorem: There exists acountable, disjoint family of cubes Qj s.t. [j �Qj ¼ A, and thus?

A

f

� �¼

1

jAj

Xj

jQjj?

Qj

f

� �¼

?�

f

� �1

jAj

Xj

jQjj ¼?�

f

� �:

Step 2. Proof of (b).By Fubini’s theorem we have f� 2 L1ð�Þ, and by stationarity we have?

��f ða; xþ yÞ dy ¼

?xþ��

f ða; yÞ dy ¼?��

f ðxa; yÞ dy ¼ f�ðxaÞ:

�

Proof of Lemma 2.23. Step 1. Proof of the first statement.W.l.o.g. we may assume that f � 0. Let � > 0. Since f�ðaÞ :¼

>�� f ða; yÞ dy 2 L1ð�Þ, we have by ð2.13Þ, and

Lemma 2.22 (a),

limR!1

?R�

f�ðxaÞ dx ¼ h f�i ¼?��

f

� �¼

?�

f

� �; ð2:48Þ

for all a 2 �0 with Pð�0Þ ¼ 1. From now on let a 2 �0 be fixed. For all y 2 �� we have?

R�

f ða; xÞ dx �ðRþ �Þ

R

� �d?ðRþ�Þ�

f ða; xþ yÞ dx;

and thus applying>�� dy yields

?R�

f ða; xÞ; dx �ðRþ �Þ

R


?��

f ða; xþ yÞ dy dx:

By stationarity of f , we find that?��

f ða; xþ yÞ dy ¼?

xþ��f ða; yÞ dy ¼

?��

f ðxa; yÞ dy ¼ f�ðxaÞ;

and thus?

R�

f ða; xÞ; dx �ðRþ �Þ

R


f�ðxaÞ dx:

By a similar argument, we obtain?

R�

f ða; xÞ; dx �ðR� �Þ

R

� �d?ðR��Þ�

f�ðxaÞ dx:

Thanks to ð2.48Þ the right-hand sides of the previous two equations converge to h>�f i, and thus we conclude that

limR!1

?R�

f ða; xÞ dx ¼?�

f

� �:

Step 2. Proof of the second statement.Set Q0 :¼ fQ ¼ qþ ‘� : Q � �; q 2 Qd; ‘ 2 Q>0g, where Q denotes the field of rational numbers. By part (a) for anyQ 2 Q0, say Q ¼ qþ ‘�, we have?

�

f ða;RxÞ1QðxÞ dx ¼jQjj�j

?qþ‘�

f ða;RxÞ ¼jQjj�j

?R‘�

f ðqa; xÞ !?�

f

� �?�

1Q;

where 1Q denotes the indicator function of the set Q. The above convergence holds for all a 2 �Q with Pð�QÞ ¼ 1.Since Q0 is countable, we can find a set �0 with Pð�0Þ ¼ 1, such that the above convergence is valid for all a 2 �0,Q 2 Q0, and such that additional we have ?

R�

jf ða; xÞj2 !?�

jf j2� �

<1: ð2:49Þ

From now on let a 2 �0. We conclude by a density argument. By linearity, for any

� 2 D :¼ spanf1Q : Q 2 Q0g;

we get


?�

f ða;RxÞ�ðxÞ dx!?�

f

� �?�

�: ð2:50Þ

Since D � L2ð�Þ is dense, for any � 2 L2ð�Þ and � > 0 we can find �0 2 D s.t. k�� 0kL2ð�Þ � �, and thus

?�

jf ða;RxÞð�ðxÞ � �0ðxÞj �?�

jf ða;RxÞj2� �1

2

k�� 0kL2ð�Þ � �?

R�

jf ða; xÞj2� �1

2

: ð2:51Þ

By the triangle inequality we have?�

f ða;RxÞ�ðxÞ dx�?�

f

� �?�

�

��

�?�

jf ða;RxÞð�ðxÞ � �0ðxÞÞj dxþ?�

f ða;RxÞ�0ðxÞ dx�?�

f

� �?�

�0��

��þ

?�

f

� � ?�

��?�

�0��

��

��:In view of ð2.49Þ, ð2.50Þ and ð2.51Þ, we deduce that

lim supR!1

?�

f ða;RxÞ�ðxÞ dx�?�

f

� �?�

�

�� 2�hk fk2L2ð�Þi

12 :

Since � > 0 is arbitrary, the claim follows. �

3. Two-scale Expansion and Homogenization Error

In this section we extend Lemma 1.2 to the multidimensional, stochastic case. Next to the corrector � we require anadditional flux corrector . It is a classical object in periodic homogenization, e.g., see [21]. In the stochastic case it hasbeen recently introduced in [12].

Proposition 3.1 (extended corrector). Suppose Assumption (S) is satisfied. For i ¼ 1; . . . ; d there exists a uniquetriple ð�i; i; qiÞ such that

(a) �i is a scalar field, i ¼ ijk is a matrix field, and �i; ijk 2 H, see Lemma (2.29).(b) P-a.s. we have

�r � aðr�i þ eiÞ ¼ 0

qi :¼ aðr�i þ eiÞ � ahomei

�4ijk ¼ @jqik � @kqijin D0ðRdÞ.

(c) i is skew symmetric, and

�r � i ¼ qi in D0ðRdÞ

with the convention that ðr � iÞj ¼Pd

k¼1 @kijk.

Sketch of the argument. The existence of �i and its properties are already explained in Proposition 2.15. The existenceargument for i is similar and omitted here. We only sketch the argument for property (c). To that end apply 4 to�r � i; then (in a distributional sense):

�4ðr � iÞj ¼ �@k4ijk ¼ @kð@jqik � @kqijÞ ¼ @jðr � qiÞ � 4qij¼ �4qij:

Hence � :¼ ðr � iÞj þ qij is harmonic in Rd, and satisfies lim supR!1>BRj�j2 <1. We argue that � is equal to a

constant by appealing to the following Liouville property: Fix x 2 Rd and consider for R > jxj the annulus SR :¼BRþjxjð0Þ n BR�jxjð0Þ. Then the mean-value property for harmonic functions, and the Cauchy–Schwarz inequaltiyimplies

j�ðxÞ � �ð0Þj ¼?

BRðxÞ� �

?BRð0Þ

�

�� 1

jBRð0Þj

ZSR

j�j

�ðRþ jxjÞd

Rd

?BRþjxjð0Þ

j�j2� �1

2 jSRjjBRþjxjð0Þj

� �12

:

As R!1, the first factor in this product remains bounded, while the second factor vanishes, since jSRj � CRd�1 forsome constant C only depending on d and jxj. Thus �ðxÞ ¼ �ð0Þ and we conclude that � is constant. Since hqiji ¼hðr � iÞji ¼ 0, we conclude that � ¼ 0 and thus �r � i ¼ qi. �

28 NEUKAMM

Remark 3.2. In dimension d ¼ 1, we simply have ¼ 0.

Theorem 3.3. Suppose P satisfies Assumption (S). Let � > 0 and f 2 L2ðRdÞ. For a 2 � let u"ða; �Þ; u0 2 H1ðRdÞdenote the unique weak solutions to

�u" �r � aðx"Þru"ða; �Þ ¼ f ;

�u0 � r � ahomru0 ¼ f :

Let ð�; Þ ¼ ð�1; . . . ; �d; 1; . . . ; dÞ denote the extended corrector of Proposition 3.1, and consider the two-scaleexpansion

Z"ða; �Þ ¼ u"ða; �Þ � u0 þ "Xdi¼1

�iða; �"Þ@iu0

!:

Then P-a.s. we have

�

ZjZ"j2 þ �

ZjrZ"j2

� Cðd; �Þ"2 �

Zj�ð�

"Þj2jru0j2 þ

Zðjð�

"Þj2 þ jað�

"Þj2j�ð�

"Þj2Þjrru0j2

� �:

Proof. By a density argument, we may assume that u0 is smooth.Step 1. Shorthand: a" :¼ að�"Þ, qi;"ðxÞ ¼ qiðx"Þ, q"ðxÞ ¼ ðq1;"ðxÞ; . . . ; qd;"ðxÞÞ, and note that

q" ¼ a"ðI þr�ð�"Þ þ IÞ � ahom:

We compute (using Einstein’s summation convention),

rZ" ¼ ru" � ðru0 þ r�ið�"Þ@iu0Þ � "�ið�"Þr@iu0

¼ ru" � ðI þ r�ð�"ÞÞru0 � "�ið�"Þr@iu0:

Then (with a" :¼ að�"Þ)

a"rZ" ¼ a"ru" � ahomru0 � q"ru0 � "�ið�"Þa"r@iu0:

Thus,

�

ZZ2" þ

ZrZ" � a"rZ" ¼ �

ZZ"ðu" � u0Þ þ

ZrZ" � a"ru" � rZ" � ahomru0|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}¼0

� �"Z�ið�"Þ@iu0Z"

�Z

q"ru0 � rZ"

� "Z�ið�"Það

�"Þr@iu0 � rZ";

ð3:1Þ

so we have three error terms.Step 2. We now discuss the second error term of ð3.1Þ. Claim:

8v 2 H1ðRdÞ :Z

q"ru0 � rv ¼ "Zr@iu0 � irv:

Indeed, in the sense of distribution we have

�r � ðqi;"@iu0Þ ¼ � ðr � qi;"Þ|fflfflfflffl{zfflfflfflffl}¼0

@iu0 � qi;" � rð@iu0Þ:

Thanks to Proposition 3.1 (c) we have with i;" ¼ ið�"Þ,

qi;" ¼ qið�"Þ ¼ �ðr � iÞð�"Þ ¼ �"r � i;":

Therefore,

�qi;" � r@iu0 ¼ "ðr � i;"Þ � r@iu0

¼ "@kijk;"@j@iu0

¼ "@kðijk;"@j@iu0Þ � "ijk;"@2jk@iu0|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}¼0

;


where we have used that ijk;" is skew-symmetric and @2jk@iu0 is symmetric (w.r.t. jk). Overall we have (in adistributional sense)

�r � ðqi;"@iu0Þ ¼ "@kðijk;"@j@iu0Þ¼ �"@kðikj;"@j@iu0Þ¼ �"r � ði;"r@iu0Þ:

Now the claim follows by testing with v.Step 3. Conclusion.For the first error term in ð3.1Þ we have,

�"

Z�ið�"Þ@iu0Z" � 2�"2

Zj�ð�

"Þj2jru0j2 þ �

2

ZZ2"|fflfflffl{zfflfflffl}

to be absorbed

:

For the second error term in ð3.1Þ, by Step 2 we have

�Z

q"ru0 � rZ" � "Zjð�

"Þjjr2u0jjrZ"j �

3

�"2Z�"

� ��2jr2u0j2 þ

�

3

ZjrZ"j2:

The third error term in ð3.1Þ is estimated as

�"Z�ið�"Það

�"Þr@iu0 � rZ" �

Z ffiffiffi3�

q"j�ð�"Þjjað

�"Þjjrru0j

ffiffiffi�3

qjrZ"j

� 3�"

2

Zj�ð�"Þj

2jað�"Þj2jrru0j2 þ �

3

ZjrZ"j2:

All together we get

�2

ZZ2" þ

�3

ZjrZ"j2 � "2

2�

Zj�ð�

"Þj2jrru0j2

þ 6�

Zðj�ð�"Þj

2jað�"Þj2 þ jð�"Þj

2Þjrru0j

!:

�

The estimate of Theorem 3.3 reveals that the rate of convergence in the homogenization limit is encoded in thebehavior of the correctors ð�; Þ. In the periodic, scalar case it is relatively easy to conclude that the two-scaleexpansion satisfies the estimate kZ"kH1ðRdÞ � Cðd; �Þ"k f kL2ðRdÞ (which is optimal in terms of scaling in "). The reason isthat in the periodic case, say when P concentrates on a 1-periodic coefficient field (and its translations), we have byPoincare’s inequality on �, the estimate

8R � 1 :

?R�

jð�; Þj2 � CðdÞ?

R�

jrð�; Þj2 � Cðd; �Þ (P-a.s.);

which combined with a Moser iteration yields

kð�; ÞkL1ðRdÞ � Cðd; �Þ:

Thus Theorem 3.3 turns into the estimate

�

ZjZ"j2 þ �

ZjrZ"j2 � Cðd; �Þ"2

Zf 2:

In the stochastic case we do not expect such a behavior. The sublinearity of ð�; Þ only yields (for u0 sufficientlyregular),

"2ZRd

ðjð�"Þj2 þ jað�

"Þj2j�ð�

"Þj2Þjrru0j2 ! 0 P-a.s.;

and we do not expect a rate for the convergence in the general ergodic case. It turns out that we need to strengthen andquantify the assumption of ergodicity in order to see a rate in the convergence above. In fact, there is a subtle interplaybetween the space dimension d, the strength of the mixing condition, and the rate of convergence. In particular, ind ¼ 2, even under the strongest quantitative ergodicity assumptions, e.g., for coefficients with finite range of

dependence, the rate is "ffiffiffiffiffiffiffiffiffiffilog 1

"

p(and thus worse than in the periodic case). This has first been observed in [14] where a

discrete situation is studied. We refer to the recent work [12], where the interplay of correlation and the decay in thetwo-scale expansion is discussed in full generality.

30 NEUKAMM

4. Quantitative Stochastic Homogenization

The goal in this section is to establish moment bounds for the correctors ð�; Þ, i.e., estimates on h>�þxj�j2 þ jj2i

that capture the optimal growth in x 2 Rd. The sublinearity of ð�; Þ yields only the behavior 1jxj2 h

>�þxj�j2 þ jj2i ! 0

for jxj ! 1, but not a quantitative growth rate. This is in contrast to the periodic case, where>�þxj�j2 þ jj2 is

bounded uniformly in x 2 Rd — a consequence of Poincare’s inequality on the unit cell of periodicity. It turns out thatin order to obtain a quantitative growth rate, we need to strengthen and quantify the assumption of ergodicity. Inparticular, we shall see that for d � 3 we obtain an estimate that is uniform jxj and for d ¼ 2 a logarithmic growth rate,provided P satisfies a strong quantitative form of ergodicity. Combined with the two-scale expansion Theorem 3.3 suchmoment bounds yield error estimates for the homogenization error. Moreover, moment bounds on the corrector are atthe basis to prove various quantitative results in stochastic homogenization, e.g., estimates on the approximation errorof ahom by representative volume elements of finite size, e.g., see [4, 12–14, 16, 17].

In the following we work in a discrete framework, i.e., Rd is replaced by Zd and the elliptic operator �r � ðarÞ isreplaced by an elliptic finite difference operator, r�ðarÞ. We do this for several reasons:

. it is easy to define model problems of random coefficients satisfying a quantitative ergodicity assumption (e.g.,i.i.d. coefficients),

. some technicalities disappear: e.g., questions of regularity on small scales,

. on the other hand: main difficulties are already present in full strength in the discrete case,

. main concepts and results naturally extend to the continuum case,

. the discrete framework is a natural setup in statistical mechanics and probability theory (e.g., random conductancemodels, see [7, 22] for recent reviews).

4.1 The discrete framework and the discrete corrector

We consider functions defined on the lattice Zd and set for 1 � p <1,

‘p :¼ f : Zd ! R : k fk‘ p :¼Xx2Zd

jf ðxÞjp !1

p

<1

8<:

9=;;

and

‘1 :¼ f f : Zd ! R : kfk‘1 :¼ supx2Zd

jf ðxÞj <1g:

Discrete calculus. Given a scalar field f : Zd ! R, and a vector field F ¼ ðF1; . . . ;FdÞ : Zd ! Rd , we set

ri f ðxÞ :¼ f ðxþ eiÞ � f ðxÞ; r�i f ðxÞ :¼ f ðx� eiÞ � f ðxÞ;

r f ¼ ðr1 f ; . . . ;rd f Þ; r�F ¼Xdi¼1

r�i Fi :

It is easy to check that for f 2 ‘p and Fi 2 ‘q (with p; q dual exponents) the integration by parts formulaXx2Zd

f ðxÞr�FðxÞ ¼Xx2Zd

r f ðxÞ � FðxÞ;

holds. Thus r� is the adjoint of r, and the discrete analogue to �r�.

Discrete elliptic operator and Green’s function. Recall that � 2 ð0; 1Þ (the ellipticity ratio) is fixed. Define

�0 :¼ fa 2 Rdd : a ¼ diagða1; . . . ; adÞ with ai 2 ð�; 1Þg � Rdd;

� :¼ fa : Zd ! �0g ¼ �Zd

:

Then for any a 2 � (and any 1 � p � 1), r�ðarÞ : ‘p! ‘p is a bounded linear operator which is uniformly ellipticand satisfies a maximum principle. We denote the Green’s function associated with r�ðarÞ by Gða; x; yÞ, i.e., Gða; �; yÞ :Zd ! R is the unique sublinear solution (resp. bounded solution if d > 2) to

r�ðarGða; �; yÞÞ ¼ �ð� � yÞ in Zd ;

where � : Zd ! f0; 1g denotes the Dirac function centered at 0.

Random coefficients. Let P denote a probability measure on ð�;�ZdBðRddÞÞ. We introduce the shift-operator

: Zd �! �; za :¼ að� þ zÞ ð4:1Þ

and always assume stationarity of P, i.e., for any z 2 Zd the mapping


8z 2 Zd : z : �! � preserves the measure P: ðD1Þ

We say P is ergodic, if

A � � is shift invariant ) PðAÞ 2 f0; 1g: ðD2Þ

Birkhoff’s ergodic theorem then implies:

limR!1

R�dX

x2R�\ZÞdf ðxaÞ ¼ h f i

for a.e. a and all f 2 L1ð�Þ.

Stationary random fields and the ‘‘horizontal’’ differential calculus. We say a function u : � Zd ! R is arandom field, if uð�; xÞ is measurable for all x 2 Zd. We say that u is a stationary random field, if

uða; xÞ ¼ uðxa; 0Þ for all x 2 Zd and P-a.e. a 2 �:

For a stationary random field u the value of huðxÞi is independent of x 2 Zd, and thus we simply write hui. We considerthe space

S :¼ fu : � Zd ! R : u is stationary and hjuj2i <1g;

which with the inner product ðu; vÞS :¼ huvi is a Hilbert space. For a random variable u we set uða; xÞ :¼ uðxaÞ. Wecall u the stationary extension of u, and note that the map

ð�Þ : L2ð�Þ ! S; u 7! u

is a linear isometric isomorphism (thanks to the stationarity of P). Note that for any u 2 S, we have

riuða; xÞ ¼ uða; xþ eiÞ � uða; xÞ ¼ uðeia; xÞ � uða; xÞ:

Motivated by this, we define for a random variable u : �! R and a random vector F : �! R the ‘‘horizontal’’derivatives

DiuðaÞ :¼ uðeiaÞ � uðaÞ; D�i uðaÞ :¼ uð�eiaÞ � uðaÞ;

Du ¼ ðD1 f ; . . . ;Dd f Þ; D�F ¼Xdi¼1

D�i Fi;ð4:2Þ

and note that we have

ruða; xÞ ¼ ðDuÞða; xÞ; r�Fða; xÞ ¼ ðD�FÞða; xÞ:

Moreover, for a random variable u 2 Lpð�Þ and a random vector F 2 Lpð�;RdÞ the integration by parts formula

huD�Fi ¼ hDu � Fi

holds, as a simple consequence of the stationarity of P.

Homogenization result in the discrete case. As in the continuum case, homogenization in the random, discrete caserelies on the notion of correctors. The correctors belong to the space

H :¼ fu : � Zd ! R : uð�; xÞ is measurable for all x 2 Zd;

ru is stationary, hjruj2i <1, and hrui ¼ 0g;

which equipped with

ðu; vÞH0:¼ hru � rvi

is a Hilbert space. Note that since ru and rv are stationary, the value of hruðxÞ � rvðxÞi does not depend on x 2 Zd,and thus we simply write hru � rvi. The following result is the discrete analogue to Proposition 2.15:

Proposition 4.1. Assume (D1) and (D2). For i ¼ 1; . . . ; d there exist unique random fields �i, qi and i ¼ ijk suchthat

(a) �i is a random scalar field, i ¼ ijk is a random matrix field, and �i; ijk 2 H.(b) P-a.s. we have

r�ðaðr�i þ eiÞÞ ¼ 0 in Zd; ð4:3Þqi ¼ aðr�i þ eiÞ � ahomei in Zd; ð4:4Þ

r�rijk ¼ rkqij �rjqik in Zd; ð4:5Þwhere ahomei :¼ haðr�i þ eiÞi, and qij denotes the jth component of the vector qi.

32 NEUKAMM

(c) i is skew symmetric, and

r�i ¼ qi in Zd;

where ðr�iÞj ¼Pd

k¼1 r�kijk.

Since the proof of the proposition is similar to the continuum case, we omit it here and refer to [1, 5, 11]. With help ofProposition 4.1 we obtain the following discrete (rescaled) analogue to Theorem 3.3:

Theorem 4.2 (Discrete two-scale expansion). Assume (D1) and (D2). Let � > 0 and f 2 L2ðZdÞ. For a 2 � letuða; �Þ; u0 : Zd ! R denote the unique square summable solutions to

�uða; �Þ þ r�ðaruða; �ÞÞ ¼ f in Zd;

�u0 þr�ðahomru0Þ ¼ f in Zd:

Let ð�; Þ ¼ ð�1; . . . ; �d; 1; . . . ; dÞ denote the extended corrector of Proposition 4.1, and consider the two-scaleexpansion

Zða; �Þ ¼ uða; �Þ � u0 þXdi¼1

�iða; �Þriu0

!:

Then for P-a.e. a 2 � we haveXx2Zd

�jZða; xÞj2 þ �jrZða; xÞj2

� Cðd; �Þ �Xx2Zd

j�ða; xÞj2jru0ðxÞj2 þXx2Zd

ðjða; xÞj2 þ jaðxÞj2j�ða; xÞj2Þjrru0ðxÞj2 !

;

where ðrru0Þij ¼ rirju0.

The statement should be compared with a rescaled (i.e., x" x) version of Theorem 3.3. The proof is (up to minor

modification regarding the transition to the discrete setting) similar to the continuum case. We omit it here and refer to[5, Proof of Proposition 3].

In the rest of this section we are interested in proving bounds for the correctors ð�; Þ.

Heuristics. To get an idea of what we can expect regarding an estimate on hj�ðxÞj2i, we consider the simplifiedequation

r�r� ¼ r�ða�Þ:

Since the divergence does not see constants, we may assume without loss of generality that haðxÞi ¼ 0. Formally asolution can be represented with help of the Green’s function GðxÞ :¼ Gðid; x; 0Þ associated with r�r:

�ðxÞ ¼Xy2Zd

Gðx� yÞr�ðaðyÞ�Þ ¼Xy2Zd

rGðx� yÞ � ðaðyÞ�Þ:

Thus

h�ðxÞ2i ¼Xy

Xy0

riGðx� yÞrjGðx� y0ÞhðaðyÞ�Þiðaðy0Þ�Þji

¼Xy

Xy0

riGðx� yÞrjGðx� y0Þhðað0Þ�Þiðaðy0 � yÞ�Þji

¼z¼y0�y X

y

Xz

riGðx� yÞrjGðx� y� zÞhðað0Þ�ÞiðaðzÞ�Þji

¼y x�y Xy

Xz

riGðyÞrjGðy� zÞhðað0Þ�ÞiðaðzÞ�Þji:

Specify to � ¼ e�, by diagonality have ðað0Þ�Þi ¼ ��ia�. Since hai ¼ 0, we arrive at

h�ðxÞ2i ¼Xy

Xz

r�GðyÞr�Gðy� zÞCðzÞ

�Xy

Xz

ðjyj þ 1Þ1�dðjy� zj þ 1Þ1�djCðzÞj;

where CðzÞ :¼ COVða�ð0Þ; a�ðzÞÞ. Note that the behavior jCðzÞj ! 0 for z!1 encodes a decay of correlations. Let usimpose the strongest possible assumption, namely independence, i.e., CðzÞ � �ðzÞ. We get


Xy

Xz

ðjyj þ 1Þ1�dðjy� zj þ 1Þ1�djCðzÞj ¼Xy

ðjyj þ 1Þ2ð1�dÞ;

and see that the right-hand side is finite if and only if d � 3.This suggests:. We can only expect moment bounds on � (uniformly in x) for d � 3.. We need assumptions on the decay of correlations of the random coefficients ( quantification of ergodicity). Regularity theory for elliptic equations is required, e.g., estimates on the gradient of the Green’s function.

4.2 Quantification of ergodicity via Spectral Gap

In this section we discuss how ergodicity can be quantified by means of a spectral gap estimate. The presentationclosely follows [15], which is an extended preprint to [13].

Definition 4.3 (vertical derivative and Spectral Gap (SG)).. For f 2 L1ð�Þ and x 2 Zd we define the vertical derivative as

@x f :¼ f � h f j F xi;

where h�jF xðaÞi denotes the conditional expectation where we condition on the -algebra F x :¼ ð�z : z 6¼ xÞ,�za :¼ aðzÞ.

. We say P satisfies (SG) with constant � > 0, if for any f 2 L2ð�Þ we have

hð f � h f iÞ2i �1

�

Xx2Zd

hð@x f Þ2i:

We might interpret the vertical derivative as follows: h�jF xi denotes the conditional expectation, where we condition onthe event that we know the value of aðzÞ for all sites z 6¼ x; thus, @x f ‘‘measures’’ how sensitive f ðaÞ reacts to changes ofthe value of a at x. The estimate (SG) is also called ‘‘Efron-Stein inequality’’ and is an example of a concentrationinequality. We refer to [24] for a review on concentration inequalities. Note that we can bound j@x f ðaÞj from above byappealing to the classical partial derivative:

j@x f ðaÞj � supf f ðaÞ � f ð ~aÞ : ~a 2 � with a ¼ ~a on Zd n fxgg �Z 1

�

@ f ðaÞ@aðxÞ

�� dx:

Let us anticipate that below in Sect. 4.3 we replace the vertical derivative @x f by a Lipschitz derivative, which isstronger than the vertical derivative and thus yields a weaker condition. Concentration inequalities such as (SG) yield anatural way to quantify ergodicity for random coefficients that rely on a product structure:

Lemma 4.4. Suppose that P is independent and identically distributed, i.e.,

P ¼ �x2ZdP0ðdxÞ for some P0 probability measure on �0:

Then h�i satisfies (SG) with constant � ¼ 1.

Proof of Lemma 4.4. The argument is standard. We follow [15] and start with preparatory remarks.. Let x1; x2; x3; . . . denote an enumeration of Zd,. Since P is a product measure, we have

h�jF xni ¼Z

�0

� P0ðdxnÞ:

. We introduce the shorthand

h�in :¼Zð�0Þn

�Yni¼1

P0ðdxiÞ;

�n :¼ h�in;�0 :¼ �;

i.e., �n does not depend on the values of aðx1Þ; . . . ; aðxnÞ.Thanks to the product structure of P, we have

34 NEUKAMM

hj@xn�j2i ¼ hhj@xn�j

2in�1i ¼ � �Z

�0

�P0ðdxnÞ��

��2* +

n�1

* +

�Jensen

� �Z

�0

�P0ðdxnÞ� �

n�1

��2* +

¼ hj�n�1 � �nj2i:

Now, the statement follows from the Martingale decomposition

hð� � h�iÞ2i ¼X1n¼1

hð�n�1 � �nÞ2i: ð4:6Þ

Here comes the argument for ð4.6Þ: Since @xh�i ¼ 0, it suffices to consider � 2 L2ð�Þ with h�i ¼ 0. By a densityargument, it suffices to consider � 2 L2ð�Þ that only depend on a finite number of coefficients, i.e., �N ¼ h�i for someN 2 N. Hence, by definition we have �0 ¼ � and �N ¼ h�i ¼ 0 for N large enough, and thus (by telescopic sum)

� ¼XNn¼1

�n�1 � �n: ð4:7Þ

Taking the square and the expected value yields

h�2i ¼XNn¼1

XNm¼1

hð�n�1 � �nÞð�m�1 � �mÞi:

Hence, ð4.6Þ follows, provided that the random variables f�n�1 � �ngn2N are independent (i.e., pairwise orthogonal inL2ð�Þ). For the argument let m > n. Since by construction �m�1 � �m does not depend on aðy1Þ; . . . ; aðym�1Þ we have

�m�1 � �m ¼ h�m�1 � �mim�1; ð4:8Þ

and since m� 1 � n, we have

h�n�1 � �nim�1 ¼ hh�in�1im�1 � hh�inim�1 ¼ h�im�1 � h�im�1 ¼ 0: ð4:9Þ

Hence, using the general identity hhuim�1vi ¼ huhvim�1i, get

hð�m�1 � �mÞð�n�1 � �nÞi ¼(4.8) hh�m�1 � �mim�1ð�n�1 � �nÞi¼ hð�m�1 � �mÞh�n�1 � �nim�1i

¼(4.9)0

and the claim follows. �

We next illustrate that (SG) not only implies, but also quantifies ergodicity. For this reason let pðt; xÞ denote theGreen’s function for the heat equation @t þ r�r (i.e., the unique function in Cð½0;1Þ; ‘2ðZdÞÞ \ C1ð�; ‘2ðZdÞÞsatisfying @t pþ r�rp ¼ 0 on ð0;1Þ Zd and pð0; xÞ ¼ �ðxÞ). Note that pðt; xÞ (which is also referred to as the heatkernel of the simple random walk on Zd) is non-negative, normalized

Px2Zd pðt; xÞ ¼ 1, and in particular, it satisfies the

on-diagonal heat kernel estimate Xx2Zd

p2ðt; xÞ � CðdÞðt þ 1Þ�d2 :

With help of pðt; xÞ we might define a semigroup ðPðtÞÞt�0 on L2ð�Þ by setting

PðtÞ : L2ð�Þ ! L2ð�Þ; PðtÞ� :¼Xx2Zd

pðt; xÞ�ða; xÞ;

where �ða; xÞ :¼ �ðxaÞ denotes the stationary extension. Thanks to the interplay of ð�Þ and r, stationarity of P impliesthat the generator of ðPtÞt�0 is given by �D�D, where D denotes the horizontal derivative, see ð4.2Þ. We thus mayequivalently write Pt� ¼ expð�tD�DÞ�. Note that the exponential is unambiguously defined, since �D�D : L2ð�Þ !L2ð�Þ is a bounded linear operator by the definition of D�D and the triangle inequality. It turns out that ergodicity canbe characterized with help of Pt.

Lemma 4.5 (Characterization and quantification of ergodicity). Let P be stationary. Consider the semigroup definedby

PðtÞ� :¼ expð�tD�DÞ�:

Then


(a) P is ergodic, if and only if

8� 2 L2ð�Þ : limt!1hjPðtÞ� � h�ij2i ¼ 0:

(b) If P satisfies (SG) with constant � > 0, then

hjPðtÞ� � h�ij2i12 �

CðdÞffiffiffi�p ðt þ 1Þ�

d4

Xx2Zd

hj@x�j2i12;

Proof of Lemma 4.5 (a). We follow the argument in [15], and consider the space of shift-invariant functions

Ið�Þ :¼ f� 2 L2ð�Þ : D� ¼ 0g;

and note that by definition, P is ergodic, if and only if Ið�Þ ¼ R. (Indeed, this can be seen by considering first indicatorfunctions of measurable sets, and then appealing to the fact that the linear span of indicator functions is dense in L2ð�Þ).Step 1. Claim:

Ið�Þ ¼ f� 2 L2ð�Þ : D�D� ¼ 0g ¼ kernel of D�D:

The inclusion � is trivial. Let � 2 L2ð�Þ satisfy D�D� ¼ 0. Then

0 ¼ h�D�D�i ¼ hjD�j2i;

and thus � 2 Ið�Þ.Step 2. Claim:

Ið�Þ? ¼ fD�F : F 2 L2ð�Þdg (in L2ð�Þ):

Since Ið�Þ is closed, it suffices to prove

ðaÞ X :¼ fD�F : F 2 L2ð�Þdg � Ið�Þ? and ðbÞ X? � Ið�Þ:

Argument for (a):

8F 2 L2ð�Þd; � 2 Ið�Þ : 0 ¼ hF � D�i ¼ hðD�FÞ�i:

Argument for (b): Let � 2 X?. Then

8F 2 L2ð�Þd : 0 ¼ h�D�Fi ¼ hD� � Fi ) � 2 Ið�Þ:

Step 3. (A priori estimates).Let � 2 L2ð�Þ and set uðtÞ :¼ PðtÞ�. Claim:

8t � 0 : hjuðtÞj2i � h�2i; ð4:10Þlimt"1hjDuðtÞj2i ¼ 0: ð4:11Þ

Recall that @tuþ D�Du ¼ 0 and uð0Þ ¼ �. Testing with uðtÞ and DuðtÞ yields

1

2

d

dthuðtÞ2i ¼

d

dtuðtÞ uðtÞ

� �¼ �hjDuðtÞj2i � 0;

1

2

d

dthjDuðtÞj2i ¼

d

dtDuðtÞ � DuðtÞ

� �¼

d

dtuðtÞD�DuðtÞ

� �¼ �hjD�DuðtÞj2i � 0:

Integration of the first identity yields ð4.10Þ andR10hjDuðtÞj2idt � h�2i <1, and thus ð4.11Þ, since t 7! hjDuðtÞj2i is

monotone (non-increasing) by the second estimate.Step 4. (Conclusion).Let � 2 L2ð�Þ and write � ¼ �0 þ �00 with �0 2 Ið�Þ? and �00 2 Ið�Þ. We claim that

PðtÞ�! �00 in L2ð�Þ as t!1: ð4:12Þ

With ð4.12Þ we can conclude the proof: If P is ergodic, then Ið�Þ ¼ R and �00 ¼ h�i. On the other hand, if PðtÞ�! h�i,then �00 ¼ h�i. Since this is true for any � and �00 is the projection onto Ið�Þ, we get Ið�Þ ¼ R.

Argument for ð4.12Þ: Since Ið�Þ is the kernel of D�D, we have PðtÞ� ¼ PðtÞ�0 þ �00, and it suffices to provePðtÞ�0 ! 0 for all �0 2 Ið�Þ?. By Step 2 for any > 0 we can find F 2 L2ð�Þd with hj�0 � D�Fj2i � . ð4.10Þ yields

8t 2 Rþ hjPðtÞð�0 � D�FÞj2i � : ð4:13Þ

We claim that

limt"1hjPðtÞD�Fj2i ¼ 0: ð4:14Þ

36 NEUKAMM

Estimate ð4.14Þ can be seen as follows. Since the shift operators e1; . . . ; ed commute, we get

PðtÞD�F ¼ expð�tD�DÞD�F ¼Xdi¼1

D�i expð�tD�DÞFi:

Hence,

hjPðtÞD�Fj2i � dXdi¼1

hjD�i PðtÞFij2i

¼stationaritydXdi¼1

hjDiPðtÞFij2i:

Now, ð4.11Þ implies ð4.14Þ. Since > 0 arbitrary, the conclusion follows. �

Proof of Lemma 4.5 (b). We follow the argument in [15].Step 1. W.l.o.g. assume that h�i ¼ 0. Set uðtÞ :¼ PðtÞ�, and recall that

uðtÞ ¼Xz2Zd

Gðt; zÞ�ðzÞ:

We have

huðtÞi ¼Xz2Zd

Gðt; zÞh�ðzÞi ¼ 0:

Thus, we can apply (SG) and obtain

hu2ðtÞi �1

�

Xy2Zd

hj@yuðtÞj2i: ð4:15Þ

We have

@yuðtÞ ¼Xz2Zd

Gðt; zÞ@yuðt; zÞ

¼Xz2Zd

Gðt; zÞ@y�zuðt; zÞ:

The combination of both yields

Xy2Zd

Xz2Zd

Gðt; zÞ@y�zuðt; zÞ

0@

1A2* +0

@1A

12

¼Xy2Zd

Xx2Zd

Gðt; y� xÞ@x�ðy� xÞ

!2* +0@

1A

12

�

4-inequality

in ðP

y2Zd hð�Þ2iÞ12 Xx2Zd

Xy2Zd

hðGðt; y� xÞ@x�ðy� xÞÞ2i

0@

1A

12

¼G is deterministic;

stationarityXx2Zd

Xy2Zd

G2ðt; y� xÞhj@x�j2i

0@

1A

12

¼Xx2Zd

hj@x�j2i12

Xy2Zd

G2ðt; y� xÞ

0@

1A

12

:

We conclude by appealing to the on-diagonal heat kernel estimateXy

G2ðt; yÞ ¼ Gð2t; 0Þ � CðdÞðt þ 1Þ�d2 :

�

The estimate in part (b) of Lemma 4.5 extends to the semigroup expð�D�ðað0ÞDÞÞ. The extension is non-trivial, sinceon the one hand, the operator r�ðað0ÞrÞ and @x do not commute, and secondly, the regularity for r�ðað0ÞrÞ is moreinvolved than that for the discrete Laplacian r�r. In [13] we obtained the following decay estimate:


Theorem 4.6 (see [13]). Let P be stationary and satisfy (SG) with constant � > 0. Consider the semigroup given by

PðtÞ :¼ expð�tD�ðað0ÞDÞÞ

Then for all exponents p with p0ðd; �Þ � p <1, all t � 0 and F 2 L2pð�Þd we have

hjPðtÞD�Fj2pi12p � Cðd; �; �; pÞðt þ 1Þ�ð

d4þ1

2ÞXy2Zd

hj@yFj2pi12p :

The proof of this theorem is out of the scope of this lecture. We only give some remarks: The exponent d4þ 1

2is optimal

and the improvement of 12

compared to the exponent in Lemma 4.5 is due to the fact that in Theorem 4.6 we considerinitial values in divergence form. The connection to homogenization is as follows: Set FðaÞ :¼ �að0Þei and note thatX

y2Zd

hj@yFj2pi12p ¼ hj@0Fj2pi

12p � sup

a;a02�jaiið0Þ � a0iið0Þj � 1� �:

Hence, hjPðtÞD�Fj2pi12p . ðt þ 1Þ�ð

d4þ1

2Þ. For d > 2, ðt þ 1Þ�ð

d4þ1

2Þ is integrable on Rþ, and thus

�iðaÞ :¼Z 1

0

PðtÞD�F dt 2 L2pð�Þ;

is well-defined and solves

D�að0ÞD�i ¼ D�F; i.e., D�ðað0ÞðD�i þ eiÞÞ ¼ 0:

Now it is easy to see that the stationary extension �iða; xÞ :¼ �iðxaÞ is a stationary solution to the corrector equation

r�ðaðr�i þ eiÞÞ ¼ 0 in Zd, P-a.s.;

with hj�ij2pi12p � Cðd; �; �Þ. We can also consider the function defined for d � 2 and T � 1 by

�T ¼Z 1

0

exp �t

T

� �PðtÞD�F dt:

From Theorem (4.6) we then deduce that

hj�T j2pi12p � Cðd; �; �; pÞ

log12 T d ¼ 2; p ¼ 1,

log T d ¼ 2; p > 1

1 d � 3.

8><>:

By applying D�ðað0ÞDÞ to �T , we find that 1T�T þ D�ðað0ÞðD�T þ �ÞÞ ¼ 0, and thus the stationary extension of �T is the

solution to the modified corrector equation.In [13], based on Theorem 4.6 we obtained various estimates on the corrector, its periodic approximation, and on the

periodic representative volume element approximation for ahom in the case of independent and identically distributedcoefficients. In the following section we take a slightly different approach to obtain moment bounds which does notinvoke the semigroup Pt.

4.3 Quantification of sublinearity in dimension d � 2

In this section we prove (under a strong quantitative ergodicity assumption) that (high) moments of r� and r arebounded, and we quantify the growth rate of hj�ðxÞj2i and hjðxÞj2i. The argument that we present combines the strategyof [5] (which relies on a Logarithmic Sobolev inequality to quantify ergodicity) and ideas of [12], where optimalgrowth rates for the correctors are obtained in the continuum setting and for strongly correlated coefficients. We alsorefer to [4, 18] where similar estimate (that are stronger in terms of stochastic integrability) are obtained for coefficientssatisfying a finite range of dependence condition (instead of the concentration inequality that we assume). Except forsome input from elliptic regularity theory (that we detail below), the argument that we present is self-contained. Westart by introducing our quantitative ergodicity assumption on P. Instead of the absolute value of the vertical derivative@x f , see Definition 4.3, we appeal to the ‘‘Lipschitz derivative’’

j@lip;x f ðaÞj :¼ supfjf ða0Þ � f ða00Þj : a0; a00 2 �; a ¼ a0 ¼ a00 in Zd n fxgg:

Definition 4.7 (Logarithmic Sobolev inequality (LSI)). We say P satisfies (LSI) with constant � > 0, if for anyrandom variable f we have

f 2 logf 2

h f 2i

� ��

1

2�

Xx2Zd

hj@lip;x f j2i:

The (LSI) is stronger than (SG). Indeed, (LSI) implies (SG) (with the same constant) as can be seen by expanding

38 NEUKAMM

f ¼ 1þ " f 0 in powers of ". In the context of stochastic homogenization (LSI) has been first used in [25]; see also [5],[11], and [10] for a recent review on (LSI) and further concentration inequalities in the context of stochastichomogenization.

Our main result is the following:

Theorem 4.8. Suppose P satisfies (D1) and (LSI) with constant � > 0. Let ð�i; iÞ denote the extended corrector ofProposition 4.1. Then for all p � 1 we have

hjr�j2p þ jrj2pi12p � Cðp; �; d; �Þ

and for all x 2 Zd we have

hj�ðxÞj2p þ jðxÞj2pi12p � Cðp; �; d; �Þ log

12ðjxj þ 2Þ d ¼ 2,

1 d � 3.

(

Note that the estimate is uniform x for d � 3. In that case we can find stationary extended correctors, i.e., ð�; Þ satisfyð�; Þða; xþ yÞ ¼ ð�; Þðxa; yÞ instead of the anchoring condition ð�; Þð0Þ ¼ 0. In dimension d ¼ 2 the correctorsdiverge logarithmically. The logarithm (and the exponent 1

2) is generically optimal as can be seen by studying the limit

of vanishing ellipticity contrast for independent and identically distributed coefficients.

Remark 4.9. Consider the two-scale expansion in Theorem 4.2. If we combine it Theorem 4.8, we deduce that theremainder Z of the two-scale expansion satisfies the estimate, for all p � 1,

XZd

�jZj2 þ �jrZj2 !p* + 1

2p

. �Xx2Zd

jru0ðxÞj2!dðxÞ þXx2Zd

jrru0ðxÞj2!dðxÞ

!12

;

where

!dðxÞ :¼logðjxj þ 2Þ d ¼ 2,

1 d � 3,

and . means � up to a constant that only depends on d; �; � and p. For d � 3 standard ‘2-regularity shows that theright-hand side is bounded by k fk‘2 . Likewise, for d ¼ 2, weighted ‘2-regularity shows that the right-hand side isestimated by k f ffiffiffiffiffiffi

!dp k‘2 . Overall we obtain the estimate

XZd

�jZj2 þ �jrZj2 !p* + 1

2p

.

XZd

jf j2!d

!12

:

For a comparison with Theorem 3.3 we need to pass to the scaled quantities Z" : "Zd ! R, Z"ðxÞ :¼ Zðx"Þ,ri;"Z"ðxÞ :¼ "�1ðriZÞðx"Þ, and f"ðxÞ :¼ "�2 f ðx"Þ. The previous estimate than turns into

X"Zd

�jZ"j2 þ �jrZ"j2 !p* + 1

2p

.

XZd

jf"j2!d

!12

" log12ð1"þ 2Þ d ¼ 2,

" d � 3.

(

Thus, for d ¼ 2 we obtain a different scaling in ".

A continuum version of Theorem 4.8 (with optimal stochastic integrability) has been recently obtained in [12]. In thediscrete case the result for d � 3 is a corollary of Theorem 4.6, while for d ¼ 2 the estimate seems to be new.

An important ingredient in the proof of Theorem 4.8 is input from elliptic regularity theory, that we recall in thefollowing paragraph.

Elliptic regularity theory. Our proof of Theorem 4.8 invokes three types of input from elliptic regularity theory:(a) an off-diagonal estimate for the Green’s function that relies on De Giorgi-Nash-Moser theory, see Lemma 4.11;(b) a weighted Meyer’s estimate established in [5], see Lemma 4.12 below;(c) an annealed Green’s function estimate for high moments of jrxryGðx; yÞj established in [25], see Lemma 4.14.

The proof of these estimates is beyond the scope of this lecture.

Remark 4.10. Estimates (a) and (b) are deterministic, in the sense that they hold for all a 2 �. Estimate (c), whichinvokes the expectation, has a different nature and is a first example of a large scale regularity result for ellipticoperator with stationary and ergodic coefficients. We refer to the recent work [11] where a rather complete large scaleregularity theory is developed. For a another approach to large scale regularity that is based on linear mixingconditions we refer to the works by Armstrong et al., see e.g., [4], the lecture notes [3] and the references therein.

Lemma 4.11 (Green’s function estimates, e.g., see [9, 19]). For any a 2 � the Green’s function (which is non-negative) satisfies


Gða; x; yÞ � Cðd; �Þlogðjxj þ 2Þ d ¼ 2,

ðjxj þ 1Þ2�d d > 2.

We do not present the proof of the estimate (which is classical). It can either be obtained by adapting the continuumargument in [19], or by integrating the heat kernel estimates in [9]. The second ingredient from elliptic regularity theoryis the following:

Lemma 4.12 (weighted Meyer’s estimate, see Proposition 1 in [5]). There exists q0 > 1 and �0 > 0 (only dependingon d and �) such that for any a 2 � and any v : Zd ! R and h : Zd ! Rd related by

r�ðarvÞ ¼ r�rh in Zd;

the following estimates hold:(a) For all ðq; �Þ 2 ½1; q0� ½0; �0� we haveX

x2Zd

jrvðxÞj2qðjxj þ 1Þ� � Cðd; q; �ÞXx2Zd

jrhðxÞj2qðjxj þ 1Þ�: ð4:16Þ

(b) For 1 < q � q01 and L � 2 consider the weight

!q;LðxÞ :¼ðjxj þ 1Þ2ðq�1Þ þ L2ð1�qÞðjxj þ 1Þ4ðq�1Þ d ¼ 2,

ðjxj þ 1Þ2dðq�1Þ d � 3.

Then we have X

x2Zd

jrvðxÞj2q!q;LðxÞ � Cðd; qÞXx2Zd

jrhðxÞj2q!q;LðxÞ: ð4:17Þ

For a proof see Step 1–Step 3 in the proof of Lemma 4 in [5]. The argument relies on a weighted Calderon–Zygmundestimate for r�r, see Proposition 1 in [5]. In the continuum case the estimates are classical. Note that the weight inð4.17Þ satisfies

Xx2Zd

!� 1

q�1

q;L ðxÞ

!¼ Cðd; qÞ

log L d ¼ 2,

1 d � 3.

ð4:18Þ

As a corollary we obtain a weighted estimate on the mixed second derivative of the Green’s function,

Corollary 4.13 (weighted Green’s function estimate). There exists q0 > 1 and �0 > 0 (only depending on d and �)such that for all ðq; �Þ 2 ½1; q0� ½0; �0� we have

supa2�

Xx2Zd

jrrGða; x; 0Þj2qðjxj þ 1Þ� � Cðq; �; d; �Þ: ð4:19Þ

Proof. Note that we have

r�x ðarxry;iGða; �; yÞÞ ¼ ðr�i �Þð� � yÞ:

Hence, the estimate follows from ð4.16Þ. �

Lemma 4.14 (annealed Green’s function estimate, see [25]). Suppose P satisfies (D1) and (LSI). Then for all p � 1

we have

hjrxryGða; x; yÞj2pi12p � Cðd; �; �Þðjx� yj þ 1Þ�d:

For a proof see [25].

Sensitivity estimate and proof of Theorem 4.8Lemma 4.15 (Sensitivity estimate). Suppose P satisfies (D1) and (D2). Then there exists �0 with Pð�0Þ ¼ 1 such thatfor i ¼ 1; . . . ; d, all a 2 �0 and all x 2 Zd we have

j@lip;xr�iða; yÞj � Cðd; �ÞjrrGða; y; xÞjjr�iða; xÞ þ eij:

Proof of Lemma 4.15. We define �0 as the set of all a 2 � such that equations ð4.3Þ and ð4.5Þ admit fori; j; k ¼ 1; . . . ; d, sublinearly growing (and thus unique) solutions with �iða; 0Þ ¼ 0 and ijkða; 0Þ ¼ 0. ByProposition 4.1 we have Pð�0Þ ¼ 1. Furthermore, note that ð4.5Þ can rewritten as

r�rijk ¼ r�Qijk;

where

40 NEUKAMM

Qijkða; xÞ :¼ ðqiða; xþ ejÞ � ekÞej � ðqiða; xþ ekÞ � ejÞek: ð4:20Þ

(Indeed, this follows from the identity riuðxÞ ¼ �ðr�i uÞðxþ eiÞ).Step 1. Let a 2 �0 and a0 2 �0 with a ¼ a0 in Zd n fxg. Set �a ¼ a� a0 and

��i :¼ �iða; �Þ � �iða0; �Þ; �ijk :¼ ijkða; �Þ � ijkða0; �Þ;�qi :¼ qiða; �Þ � qiða0; �Þ; �Qijk :¼ Qijkða; �Þ � Qijkða0; �Þ:

Then a direct calculation (using ð4.3Þ–ð4.5Þ, and the fact that �aðyÞ ¼ 0 for all y 6¼ x) yields

r�ðar��iÞ ¼ �r�ð�aðr�iða0; �Þ þ eiÞÞ; ð4:21Þr�r�ijk ¼ r��Qijk; ð4:22Þ�qiðyÞ ¼ �aðyÞðr�ða; xÞ þ eiÞ þ a0ðyÞr��iðyÞ ð4:23Þ�Qijk ¼ ð�aðyÞðr�ða; xþ ejÞ þ eiÞ � ekÞej ð4:24Þ

� ð�aðyÞðr�ða; xþ ekÞ þ eiÞ � ejÞekþ ða0ðyÞðr��iðyþ ejÞ � ekÞej� ða0ðyÞr��iðyþ ekÞ � ejÞek:

Since ��i and �ijk are sublinear (as differences of sublinear functions), we may test with the Green’s function and get

r��iðyÞ ¼ �ryrxGða; y; xÞ � �aðxÞðr�iða0; xÞ þ eiÞ ð4:25Þ

Applying ð4.25Þ with y ¼ x and the roles of a and a0 interchanged, yields

ðr�ða0; xÞ þ eiÞ � ðr�ða; xÞ þ eiÞ ¼ �rxrxGða0; x; xÞðr�iða; xÞ þ eiÞ;

and thus

jr�ða0; xÞ þ eij � ðjrxrxGða0; x; xÞj þ 1Þjr�iða; xÞ þ eij ð4:26Þ� ð1� þ 1Þjr�iða; xÞ þ eij:

We conclude that

j�r�iðyÞj ¼ Cðd; �ÞjryrxGða; y; xÞjjr�iða; xÞ þ eij ð4:27Þ

Step 2.We claim that a 2 �0, a0 2 � with a ¼ a0 on Zd n fxg implies that a0 2 �0. In view of the definition of �0, we need toshow existence of sublinear solutions to ð4.3Þ and ð4.5Þ for a0. Indeed, this can be inferred as follows: Equations ð4.21Þand ð4.22Þ admit unique sublinear solutions ��i and �ijk with ��ið0Þ ¼ �ijkð0Þ ¼ 0, since the right-hand side of ð4.21Þis the divergence of a compactly supported function, and the right-hand side of ð4.22Þ is the divergence of a squaresummable function. Now the sought for sublinear solutions are given by �iða0; �Þ :¼ �iða; �Þ þ ��i and ijkða0; �Þ ¼ijkða; �Þ þ �ijk. As a consequence of this stability of �0 w.r.t. compactly supported variations of a, when estimatingj@lip;x f ðaÞj for a 2 �0, we only need to take the sup (in the definition of the Lipschitz derivative) over fields a0; a00 2 �0

with a ¼ a0 ¼ a00 in Zd n fxg into account. Thus, the claimed estimate follow from ð4.27Þ. �

We combine the sensitivity estimate with the weighted Green’s function estimate, Corollary 4.13, and the followingconsequence of (LSI),

Lemma 4.16. Let P satisfy (LSI) with constant � > 0. Then for any 1 � p <1, any � > 0 and all random variablesf we have the estimates

hjf � h f ij2pi12p � Cðp; �Þ

Xx2Zd

j@lip;x f j2 !p* + 1

2p

; ð4:28Þ

hjf j2pi12p � Cð�; p; �Þhjf j2i

12 þ �

Xx2Zd

j@lip;x f j2 !p* + 1

2p

: ð4:29Þ

Estimate ð4.28Þ for p ¼ 1 is the usual Spectral Gap estimate, which is implied by (LSI). ð4.28Þ for p > 1 follows fromthe estimate for p ¼ 1 by the argument in [13]. For a proof of ð4.29Þ we refer to [25]. We are now in position toestablish moment bounds for r�i and ri:

Lemma 4.17. Suppose P satisfies (D1) and (LSI). Then for all 1 � p <1

hjr�i þ eij2p þ jrij2pi12p � Cðp; �; d; �Þ:

Proof. Step 1. Proof of the bound for r�i.


Note that we have hjr�i þ eij2i12 � Cðd; �Þ by construction. Hence, in view of Lemma 4.16 we only need to prove that

I :¼Xx2Zd

j@lip;xðr�ið0Þ þ eiÞj2 !p* + 1

2p

� Cðp; �; d; �Þhjr�i þ eij2pi12 p ;

since then, by choosing � sufficiently small, the right-hand side of the estimate in Lemma 4.16 can be absorbed into theleft-hand side. An application of Lemma 4.15 yields

I � Cðd; �ÞXx2Zd

jrrGða; 0; xÞj2jr�iða; xÞ þ eij2 !p* + 1

2p

We want to estimate the right-hand side by appealing to Corollary 4.13. To that end fix an exponent � > 0 for whichthe corollary applies, and suppose that p 1 is so large, such that �ðp� 1Þ > d and q :¼ p

p�1falls into the range of

Corollary 4.13. Then,

Xx2Zd

jrrGða; 0; xÞj2jr�iða; xÞ þ eij2 !p

�Xx2Zd

jrrGða; 0; xÞj2qðjxj þ 1Þ� !p�1 X

x2Zd

jr�iða; xÞ þ eij2pðjxj þ 1Þ��ðp�1ÞðxÞ

!

� Cðd; �; �; pÞXx2Zd

jr�iða; xÞ þ eij2pðjxj þ 1Þ��ðp�1ÞðxÞ

!:

We take the expectation, exploit stationarity, and arrive at

I � hjr�i þ eij2pi12p

Xx2Zd

ðjxj þ 1Þ��ðp�1ÞðxÞ

! 12p

:

Since �ðp� 1Þ > d, the claimed bound follows.Step 2. Proof of the bound for ri.As in the proof of Lemma 4.15 we write ð4.5Þ in the form r�rijk ¼ r�Qijk with Qijk defined in ð4.20Þ. Step 1 impliesthat the stationary random field Qijk has finite 2pth moment, and thus the ergodic theorem yields, P-a.s.

lim supL!1

L�dX

L�\Zd

jQijkj2p ! 1

2p

¼ hjQijkj2pi12p � CðdÞhjr�i þ eij2pi

12p :

We claim that rijk inherits this property, i.e.,

lim supL!1

L�dX

L�\Zd

jrijkj2p ! 1

2p

� CðdÞhjr�i þ eij2pi12p ; ð4:30Þ

which by the ergodic theorem then yields the sought for bound on hjrj2pi12p � Cðp; �; d; �Þ. Estimate ð4.30Þ can be

seen as follows: For L 1 let �L denote a cut-off function for L� in 2L�, and let L denote the unique solution withLð0Þ ¼ 0 and rL 2 ‘2 to

r�rL ¼ r�ðQijk�LÞ:

Then maximal ‘p regularity for r�r yields

L�dX

L�\Zd

jrLj2p ! 1

2p

� Cðd; pÞ L�dXZd

jQijk�Lj2p ! 1

2p

� L�dX

2L�\Zd

jQijkj2p ! 1

2p

:

We conclude that rL weakly converges locally in ‘2p to r, and thus

L�dX

L�\Zd

jrj2p ! 1

2p

� lim infL!1

L�dX

L�\Zd

jrLj2p ! 1

2p

� lim supL!1

L�dX

2L�\Zd

jQijkj2p ! 1

2p

� CðdÞhjr�i þ eij2pi12p :

Passing to the limit L!1 yields ð4.30Þ. �

42 NEUKAMM

Now we are in position to prove Theorem 4.8:

Proof of Theorem 4.8. The moment bounds for r� and r are already proven in Lemma 4.17. It remains to quantifythe growth of the extended corrector. Note that it suffices to prove the estimate for large p. We follow the idea in [12].Yet, we replace the input from large scale regularity theory by the regularity estimates discussed above. To easenotation, fix indices i; j; k ¼ 1; . . . ; d, and recall the definition of Qijk, see ð4.20Þ. We simply write e; �; ; q, and Q

instead of ei; �i; ijk; qi and Qijk. Furthermore, for convenience we use the notationRf ðxÞ dx and

>L� f ðxÞ dx forP

x2Zd f ðxÞ and 1#ðL�\ZdÞ

Px2ðL�\ZdÞ f ðxÞ, respectively. Below . denotes � up to a constant that can be chosen only

depending on d; �; �; � and p.Step 1. We claim that for any L � 2 and x 2 Zd:

ð�; ÞðxÞ �?

L�

ð�; Þðyþ xÞ dy��

��2p* + 1

2p

.log

12 L d ¼ 2,

1 d � 3.

(

Since ðr�;rÞ is stationary, it suffices to prove the estimate for x ¼ 0. Therefore consider

FðaÞ :¼ ð�; Þða; 0Þ �?

L�

ð�; Þða; yÞ dy;

which is a random variable with vanishing expectation. Hence, in view of ð4.28Þ it suffices to show

Xx2Zd

j@lip;xFj2 !p* + 1

2p

. log12 L: ð4:31Þ

Substep 1.1. Lipschitz estimate for F.We claim that for any a 2 �0 we have

j@lip;xFðaÞj .

Xx0:jx0�xj�1

jr�ða; x0Þ þ ej

!ðjrvða; xÞj þ jrhðxÞjÞ ð4:32Þ

þZjryrxGða; y; xÞjjr�ða; xÞ þ ejjrhðyÞj dy

!;

where h : Zd ! R denotes the unique sublinear solution to

r�rh ¼ ��1

#ðL� \ ZdÞ1L�\Zd subject to hð0Þ ¼ 0;

where � denotes the Dirac function centered at 0, and 1L�\Zd the indicator function for L� \ Zd, and vða; �Þ denotes theunique (sublinear) solution to

r�ðarvÞ ¼ �r�rh; vð0Þ ¼ 0:

For the argument, first note that F admits the representation

F ¼Xy2Zd

ðr�ða; yÞ;rða; yÞÞ � ðrhðyÞ;rhðyÞÞ:

Representing h with the fundamental solution to r�r shows that

jrhðxÞj � CðdÞðminfjxj þ 1;LgÞðjxj þ 1Þ�d: ð4:33Þ

Next we would like to estimate @lip;xF. In order to do so, recall the definition of �0 from Lemma 4.15, and let a; a0 2 �0

with a ¼ a0 on Zd n fxg. Let �a, ��, � and �Q be defined by ð4.21Þ–ð4.24Þ. Then

FðaÞ � Fða0Þ ¼Zðr��;r�Þ � ðrh;rhÞ

¼Z�r�� ðarvÞ þ r� � rh

¼Z�ðar��Þ � rvþ r� � rh

¼ �aðxÞðr�ða0; xÞ þ eÞ � rvðxÞ þZr�Qða; yÞ � rhðyÞ dy ¼: I þ II;

where the last step holds thanks to equations ð4.21Þ and ð4.22Þ. By ð4.26Þ, the modulus of the first term is estimated by

jIj . jr�ða; xÞ þ ejjrvða; xÞj:

Regarding II, from ð4.24Þ and ð4.27Þ, we deduce that


jIIj .�ðjr�ða; xþ ejÞ þ ej þ jr�ða; xþ ekÞ þ ejÞjrhðxÞj

þZjryrxGða; y; xÞjjr�ða; xÞ þ ejjrhðyÞj dy

�:

The combination of the previous estimates yields ð4.32Þ.Substep 1.2. Estimate of the first term in ð4.32Þ.In this step we estimate the first term on the right-hand side of ð4.32Þ. Set Hða; xÞ :¼

Px0:jx0�xj�1 jr�ða; x0Þ þ ej. We

claim that

Xx2Zd

jHða; xÞj2ðjrvða; xÞj þ jrhðxÞjÞ2 !p* + 1

2p

.log

12 L d ¼ 2,

1 d � 3.

(ð4:34Þ

For the argument we may assume that q ¼ pp�1

is sufficiently small, such that Lemma 4.12 applies. Set ! :¼ !q;L, seeð4.18Þ, and note that

Xx2Zd

jHða; xÞj2ðjrvða; xÞj þ jrhðxÞjÞ2 !p

�Xx2Zd

jHða; xÞj2p!�1

q�1ðxÞ

! Xx2Zd

ðjrvða; xÞj þ jrhðxÞjÞ2q!ðxÞ

!p�1

:

The weighted Meyer’s estimate Lemma 4.12 yieldsXx2Zd

ðjrvða; xÞj þ jrhðxÞjÞ2q!ðxÞ .Xx2Zd

jrhðxÞj2q!ðxÞ .log L d ¼ 2,

1 d � 3,

where the last estimates follows by a direct calculation using ð4.33Þ and the definition of ! ¼ !q;L. On the other hand,by Lemma 4.17 the moments of H are bounded, and since H is stationary, we deduce that

Xx2Zd

Xx2Zd

jHða; xÞj2p!�1

q�1ðxÞ

* +¼ hjHj2pi

XZd

!�1

q�1 .log L d ¼ 2,

1 d � 3,

The combination of the previous two estimates yields ð4.34Þ.Substep 1.3. Estimate for the second term in ð4.32Þ and conclusion of ð4.31Þ.Set Hða; y; xÞ :¼ jrrGða; y; xÞjjr�ða; xÞ þ ej. Then two applications of the triangle inequality, and Cauchy–Schwarzinequality in probability, yield

III :¼Xx2Zd

Xy2Zd

jHða; y; xÞjjrhðyÞj

0@

1A20

@1A

p* +1p

�Xx2Zd

Xy2Zd

jHða; y; xÞjjrhðyÞj

0@

1A2p* +1

p

¼Xx2Zd

Xy;y02Zd

jHða; y; xÞjjHða; y0; xÞjjrhðyÞjjrhðy0Þj

0@

1Ap* +1

p

�X

x;y;y02Zd

jrhðyÞjjrhðy0ÞjhjHða; y; xÞj2pi12phjHða; y0; xÞj2pi

12p

�Xx2Zd

Xy2Zd

jrhðyÞjhjHða; y; xÞj2pi1p

0@

1A2

¼Xx2Zd

Xy2Zd

jrhðyÞjhjHða; y� x; 0Þj2pi1p

0@

1A2

;

where the last identity holds thanks to the identity

Hða; y; xÞ ¼ Hðx; y; 0Þ; ð4:35Þ

44 NEUKAMM

and stationarity of P. In view of the moment bounds on r�, see Lemma 4.17, and the annealed Green’s functionestimate, see Lemma 4.14, we have

hjHða; y� x; 0Þj2pi1p . ðjx� yj þ 1Þ�2d ;

and thus we arrive at

III .Xx2Zd

Xy2Zd

jrhðyÞjðjx� yj þ 1Þ�2d

0@

1A2

. krhk2‘2 .log L d ¼ 2,

1 d � 3,

where the last two estimates hold due to Young’s convolution estimate and a direct calculation that uses ð4.33Þ.Combined with ð4.34Þ and ð4.32Þ, we eventually get ð4.31Þ.Step 2. We claim that for any L � 2 any x 2 Zd we have

?L�

ðr�;rÞðxþ yÞ dy��

��2p* + 1

2p

.L�1 log

12p L d ¼ 2,

L�d

2p�1p d � 3.

8<: ð4:36Þ

Since ðr�;rÞ is stationary, it suffices to prove the estimate for x ¼ 0. Therefore consider

F0ðaÞ :¼Zðr�;rÞða; yÞ � mLðyÞ dy; mL :¼

1

#ðL� \ ZdÞ1L�\Zd e0;

where e0 denotes an arbitrary unit vector in Rd Rd. It suffices to show that hjF0j2pi12p is bounded by the right-hand

side of ð4.36Þ. Since the expectation of F0 is zero, by ð4.28Þ we only need to show that

Xx2Zd

j@lip;xF0j2 !p* + 1

2p

.L�1 log

12p L d ¼ 2,

L�d

2p�1p d � 3.

8<: ð4:37Þ

Following the argument in Substep 1.1 (with rh replaced by mL) we obtain the estimate

j@lip;xF0ðaÞj .

Xx0:jx0�xj�1

jr�ða; x0Þ þ ej

!ðjrvða; xÞj þ jmLðxÞjÞ ð4:38Þ

þZjryrxGða; y; xÞjjr�ða; xÞ þ ejjmLðyÞj dy

!

¼: IðxÞ þ IIðxÞ; ð4:39Þwhere v denotes the unique sublinear solution to

r�ðarvÞ ¼ �r�mL; vð0Þ ¼ 0:

In order to get ð4.37Þ, suppose that p 1 is so large, such that Lemma 4.12 applies with q :¼ pp�1

. Then, withHða; xÞ :¼

Px0:jx0�xj�1 jr�ða; x0Þ þ ej, we get

Xx2Zd

jIðxÞj2 !p* + 1

2p

�Xx2Zd

jHða; xÞj2jrvðxÞj2 !p* + 1

2p

þXx2Zd

jHða; xÞj2mLðxÞ2 !p* + 1

2p

:

By Jensen’s inequality, Lemma 4.17 and the definition of mL, the second term is bounded by L�d2 , while for the first

term we appeal to Holder’s inequality and Lemma 4.12. As in Substep 1.2 we get

Xx2Zd

jHða; xÞj2jrvðxÞj2 !p* + 1

2p

.

Xx2Zd

m2qL !q;L

! 12q

log12p L d ¼ 2,

1 d � 3.

(

.L�1 log

12p L d ¼ 2,

L�d2q d � 3.

(

Likewise, the estimate in Substep 1.3, with rh replaced by mL yields

Xx2Zd

jIIðxÞj2 !p* + 1

2p

. kmLk‘2 � L�d2 :

The combination of the previous estimates yields ð4.37Þ, and thus ð4.36Þ.


Step 3.We claim that for any L � 2 and x 2 Zd we have for all p � p0 (only depending on d and �),

?L�

ð�; Þðxþ yÞ � ð�; ÞðyÞ dy��

��2p* + 1

2p

.jxjL

log12p L d ¼ 2,

1 d � 3.

(

For the argument note that there exists a path � � Zd with j�j :¼ #� . jxj and e : �! f�e1; . . . ;�edg s.t. for anyu : Zd ! R we have

uðx0Þ � uðxÞ ¼Xp2�ruðpÞ � eðpÞ:

Hence,

F00ðaÞ :¼?

L�

ð�; Þðxþ yÞ � ð�; ÞðyÞ dy ¼Xp2�

?L�

ðr�;rÞða; yþ pÞ � ðeðpÞ; eðpÞÞ dy;

and thus by the triangle inequality and stationarity of ðr�;rÞ, and the estimate of Step 2,

hjF00ðaÞj2pi12p �

Xp2�

?L�

ðr�;rÞða; yþ pÞÞ � ðeðpÞ; eðpÞÞ��

��2p* + 1

2p

¼Xp2�

?L�

ðr�;rÞða; yÞ��

��2p* + 1

2p

.jxjL

log12p L d ¼ 2,

1 d � 3,

(

where in the last step we assumed that p is so large, such that L�d

2p�1p � L�1 for d � 3.

Step 4. Conclusion.Choose L ¼ jxj þ 2. Then by the estimate in Step 1 and in Step 3,

hjð�; ÞðxÞj2pi12p ¼ hjð�; ÞðxÞ � ð�; Þð0Þj2pi

12p

� ð�; ÞðxÞ �?

L�

ð�; Þðyþ xÞ dy��

��2p* + 1

2p

þ?

L�

ð�; Þðyþ xÞ � ð�; ÞðyÞ dy��

��2p* + 1

2p

þ ð�; Þð0Þ �?

L�

ð�; ÞðyÞ dy��

��2p* + 1

2p

.log

12ðjxj þ 2Þ d ¼ 2,

1 d � 3.

(

�

A Solutions to Problem 1–5

Proof of Problem 1. For simplicity we set a" :¼ að�"Þ. By the fundamental theorem of calculus we have

u"ðxÞ � u"ð0Þ ¼Z x

0

@xu"ðx0Þ dx0 ¼Z x

0

a�1" ðx

0Þj"ðx0Þ dx0;

where j" denotes the flux

j"ðxÞ :¼ aðx"Þ@xu"ðxÞ:

From ð1.1Þ we learn that

j"ðxÞ ¼ c" �Z x

0

f ðx0Þ dx0

for a constant c" 2 R, which is uniquely determined by ð1.2Þ: Indeed, we have

46 NEUKAMM

0 ¼ u"ðLÞ � u"ð0Þ ¼Z L

0

a�1" ðx

0Þj"ðx0Þ dx0

¼Z L

0

a�1" ðx

0Þ c" �Z x0

0


dx0;

and thus,

c" ¼Z L

0

a�1" ðx

0Þ dx0� ��1Z L

0

Z x0

0

a�1" ðx

0Þ f ðx00Þ dx00 dx0:

Since u"ðLÞ ¼ 0, we get the representation, ð1.5Þ, i.e.,

u"ðxÞ ¼Z x

0

a�1" ðx

0Þ c" �Z x0

0


dx0:

Since f and a" are smooth (by assumption), the right-hand side defines a smooth solution to ð1.1Þ and ð1.2Þ. �

Proof of Problem 2. Application of Problem 1.1 yieldsZ L

0

a�1" ¼ a�1

0 Lþ Oð"Þ; where a0 ¼Z 1

0

a�1

� ��1

c" ¼ c0 þ Oð"Þ where c0 :¼ a0

? L

0

a�10

Z x0

0

f ðx00Þ dx00dx0 ¼? L

0

Z x0

0

f ðx00Þ dx00dx0;

u"ðxÞ ! u0ðxÞ þ Oð"Þ where u0ðxÞ :¼ a�10

Z x

0

c0 �Z x0

0

f ðx00Þdx00� �

dx0:

Finally, it is easy to check that u0 is smooth and solves ð1.3Þ and ð1.4Þ. �

Proof of Problem 3. Recall that u admits the representation

uðxÞ ¼Z x

0

a�1ðx0Þðc� x0Þ dx0;

for some c 2 R. Hence, u0ðxÞ ¼ c�xaðxÞ and thus

u is quadratic, u0 is affine, að�Þ is a constant:

�

Proof of Problem 4. We first notice that M0 :¼ max �O u0 ¼ 18a0

. Indeed, this follows from

u0ðxÞ ¼ a�10

Z x

0

ð12� x0Þ dx0:

We conclude by appealing to the quantitative homogenization result max �O ju" � u0j ¼ Oð"Þ:

M" � u"ð12Þ ¼ u0ð12Þ þ Oð"Þ ¼ M0 þ Oð"Þ;

and for some x" we have

M" ¼ u"ðx"Þ ¼ u0ðx"Þ þ Oð"Þ � M0 þ Oð"Þ:

Hence, M" ¼ M0 þ Oð"Þ. �

Proof of Problem 5. We argue by contradiction and assume that (for a subsequence)ZO

j@xu" � @xu0j2! 0;

which implies that @xu"ðxÞ ! @xu0ðxÞ for a.e. x 2 O for a subsequence. The representation formula and a directcomputation shows that

@xu"ðxÞ ¼ðc" � xÞa"ðxÞ

@xu0ðxÞ ¼ðc0 � xÞ

a0

Since c"! c0 (as shown in the proof of Problem 2), we deduce that 1a"ðxÞ !

1a0

for a.e. x 2 O. Combined with thedominated convergence theorem, we conclude that 1

a"! 1

a0in L2ðOÞ, and thus

ROa"!

ROa0. However, by Problem 1.1

we have ZO

a"!ZO

Z 1

0

a 6¼ZO

a0 unless a is a constant function:

The second statement is a direct consequence of an integration by parts and Problem 2. �


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48 NEUKAMM

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