Quantitative Stochastic Homogenization

The goal in this section is to establishmoment boundsfor the correctors ð; Þ, i.e., estimates onh^>þxjj²þ j j²i that capture the optimal growth inx2R^d. The sublinearity ofð; Þyields only the behavior_jxj¹2h^>þxjj²þ j j²i !0 for jxj ! 1, but not a quantitative growth rate. This is in contrast to the periodic case, where ^>þxjj²þ j j² is bounded uniformly inx2R^d— a consequence of Poincare´’s inequality on the unit cell of periodicity. It turns out that in order to obtain a quantitative growth rate, we need to strengthen and quantify the assumption of ergodicity. In particular, we shall see that ford3we obtain an estimate that is uniformjxjand ford¼2a logarithmic growth rate, providedPsatisfies a strongquantitative formof ergodicity. Combined with the two-scale expansion Theorem 3.3 such moment bounds yield error estimates for the homogenization error. Moreover, moment bounds on the corrector are at the basis to prove various quantitative results in stochastic homogenization, e.g., estimates on the approximation error of 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.,R^d is replaced byZ^d and the elliptic operatorr ðarÞis replaced by an elliptic finite difference operator, rðarÞ. 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 conductance models, see [7, 22] for recent reviews).

4.1 The discrete framework and the discrete corrector

We consider functions defined on the latticeZ^d and set for1p<1,

‘^p:¼ f :Z^d !R:kfk_‘p :¼ X

x2Z^d

jfðxÞj^p

!¹_p

<1 8<

:

9=

;; and

‘¹:¼ ff :Z^d !R:kfk_‘¹ :¼sup

x2Z^d

jfðxÞj<1g:

Discrete calculus. Given a scalar field f :Z^d !R, and a vector fieldF¼ ðF1;. . .;FdÞ:Z^d!R^d, we set rifðxÞ:¼ fðxþeiÞ fðxÞ; r_ifðxÞ:¼ fðxeiÞ fðxÞ;

rf ¼ ðr1f;. . .;rdfÞ; rF¼X^d

i¼1

r_iFi:

It is easy to check that for f 2‘^p andFi2‘^q (with p;qdual exponents) the integration byparts formula X

x2Z^d

fðxÞrFðxÞ ¼X

x2Z^d

rfðxÞ FðxÞ;

holds. Thusr is the adjoint ofr, and the discrete analogue tor.

Discrete elliptic operator and Green’s function. Recall that 2 ð0;1Þ(the ellipticity ratio) is fixed. Define 0 :¼ fa2R^dd:a¼diagða1;. . .;adÞwithai2 ð;1Þg R^dd;

:¼ fa:Z^d !₀g ¼^Zd:

Then for anya2(and any1 p 1),rðarÞ:‘^p !‘^p is a bounded linear operator which is uniformly elliptic and satisfies a maximum principle. We denote the Green’s function associated withrðarÞbyGða;x;yÞ, i.e.,Gða;;yÞ: Z^d !Ris the unique sublinear solution (resp. bounded solution ifd>2) to

rðarGða;;yÞÞ ¼ð yÞ inZ^d; where:Z^d! f0;1gdenotes the Dirac function centered at 0.

Random coefficients. Let Pdenote a probability measure onð;Z^dBðR^ddÞÞ. We introduce the shift-operator

:Z^d!; za:¼að þzÞ ð4:1Þ

and always assume stationarityofP, i.e., for anyz2Z^d the mapping

8z2Z^d: z: ! preserves the measureP: ðD1Þ We sayPisergodic, if

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

Birkhoff’s ergodic theorem then implies:

R!1lim R^d X

x2R\ZÞ^d

fðxaÞ ¼ hfi for a.e.a and all f 2L¹ðÞ.

Stationary random fields and the ‘‘horizontal’’ differential calculus. We say a function u: Z^d !R is a random field, if uð;xÞis measurable for allx2Z^d. We say thatuis a stationary random field, if

uða;xÞ ¼uðxa;0Þ for allx2Z^d andP-a.e.a2:

For a stationary random fielduthe value ofhuðxÞiis independent ofx2Z^d, and thus we simply writehui. We consider the space

S:¼ fu: Z^d!R:u is stationary andhjuj²i<1g;

which with the inner productðu;vÞ_S:¼ huvi is a Hilbert space. For a random variableuwe setuða;xÞ:¼uðxaÞ. We calluthe stationary extension ofu, and note that the map

ðÞ:L²ðÞ !S; u7!u

is a linear isometric isomorphism (thanks to the stationarity of P). Note that for anyu2S, we have riuða;xÞ ¼uða;xþe_iÞ uða;xÞ ¼uð_eia;xÞ uða;xÞ:

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

derivatives

DiuðaÞ:¼uðe_iaÞ uðaÞ; D_iuðaÞ:¼uðe_iaÞ uðaÞ;

Du¼ ðD1f;. . .;DdfÞ; DF¼X^d

i¼1

D_iFi; ð4:2Þ

and note that we have

ruða;xÞ ¼ ðDuÞða;xÞ; rFða;xÞ ¼ ðDFÞða;xÞ:

Moreover, for a random variableu2L^pðÞand a random vectorF2L^pð;R^dÞthe integration by parts formula huDFi ¼ hDuFi

holds, as a simple consequence of the stationarity ofP.

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

H:¼ fu: Z^d !R:uð;xÞis measurable for allx2Z^d;

ruis stationary,hjruj²i<1, andhrui ¼0g;

which equipped with

ðu;vÞ_H0 :¼ hru rvi

is a Hilbert space. Note that sinceruandrvare stationary, the value ofhruðxÞ rvðxÞidoes not depend onx2Z^d, and thus we simply writehru rvi. The following result is the discrete analogue to Proposition 2.15:

Proposition 4.1. Assume (D1) and (D2). Fori¼1;. . .;d there exist unique random fieldsi,qi and i¼ ijksuch that

(a) iis a random scalar field, i¼ ijkis a random matrix field, andi; ijk2H. (b) P-a.s. we have

rðaðriþeiÞÞ ¼0 inZ^d; ð4:3Þ

q_i¼aðr_iþe_iÞ a_home_i inZ^d; ð4:4Þ rr ijk¼ rkqij rjqik inZ^d; ð4:5Þ whereahomei:¼ haðriþeiÞi, andqij denotes the jth component of the vectorqi.

(c) i is skew symmetric, and

r i¼qi inZ^d; whereðr iÞ_j¼Pd

k¼1r_k ijk.

Since the proof of the proposition is similar to the continuum case, we omit it here and refer to [1, 5, 11]. With help of Proposition 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 2L²ðZ^dÞ. For a2 let uða;Þ;u₀:Z^d!Rdenote the unique square summable solutions to

uða;Þ þ rðaruða;ÞÞ ¼ f inZ^d; u₀þ rða_homru₀Þ ¼ f inZ^d:

Let ð; Þ ¼ ð₁;. . .; _d; ₁;. . .; _dÞ denote the extended corrector of Proposition 4.1, and consider the two-scale expansion

Zða;Þ ¼uða;Þ u0þX^d

i¼1

iða;Þriu0

! : Then for P-a.e.a2we have

X

x2Z^d

jZða;xÞj²þjrZða;xÞj²

Cðd; Þ X

x2Z^d

jða;xÞj²jru₀ðxÞj²þX

x2Z^d

ðj ða;xÞj²þ jaðxÞj²jða;xÞj²Þjrru₀ðxÞj²

!

; whereðrru₀Þij¼ 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 correctorsð; Þ.

Heuristics. To get an idea of what we can expect regarding an estimate on hjðxÞj²i, we consider the simplified equation

rr¼ rðaÞ:

Since the divergence does not see constants, we may assume without loss of generality that haðxÞi ¼0. Formally a solution can be represented with help of the Green’s function GðxÞ:¼Gðid;x;0Þassociated withrr:

ðxÞ ¼X

y2Z^d

GðxyÞrðaðyÞÞ ¼X

y2Z^d

rGðxyÞ ðaðyÞÞ:

Thus

hðxÞ²i ¼X

y

X

y⁰

riGðxyÞrjGðxy⁰ÞhðaðyÞÞ_iðaðy⁰ÞÞ_ji

¼X

y

X

y⁰

riGðxyÞrjGðxy⁰Þhðað0ÞÞ_iðaðy⁰yÞÞ_ji

z¼y¼⁰yX

y

X

z

riGðxyÞrjGðxyzÞhðað0ÞÞ_iðaðzÞÞ_ji

y ¼xyX

y

X

z

riGðyÞrjGðyzÞhðað0ÞÞ_iðaðzÞÞ_ji:

Specify to¼e, by diagonality haveðað0ÞÞ_i¼ia. Sincehai ¼0, we arrive at hðxÞ²i ¼X

y

X

z

rGðyÞrGðyzÞCðzÞ

X

y

X

z

ðjyj þ1Þ^1dðjyzj þ1Þ^1djCðzÞj;

whereCðzÞ:¼COVðað0Þ;aðzÞÞ. Note that the behaviorjCðzÞj !0forz! 1encodes a decay of correlations. Let us impose the strongest possible assumption, namely independence, i.e.,CðzÞ ðzÞ. We get

X

y

X

z

ðjyj þ1Þ^1dðjyzj þ1Þ^1djCðzÞj ¼X

y

ðjyj þ1Þ^2ð1dÞ; and see that the right-hand side is finite if and only ifd3.

This suggests:

. We can only expect moment bounds on(uniformly inx) ford3.

. 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 presentation closely follows [15], which is an extended preprint to [13].

Definition 4.3 (vertical derivative and Spectral Gap (SG)).

. For f 2L¹ðÞandx2Z^d we define the vertical derivative as

@xf :¼f hf jFxi;

wherehjFxðaÞi denotes the conditional expectation where we condition on the -algebra Fx:¼ ð_z:z6¼xÞ, _za:¼aðzÞ.

. We sayPsatisfies (SG) with constant >0, if for any f 2L²ðÞwe have hðf hfiÞ²i 1

X

x2Z^d

hð@_xfÞ²i:

We might interpret the vertical derivative as follows:hjFxidenotes the conditional expectation, where we condition on the event that we know the value ofaðzÞfor all sitesz6¼x; thus,@_xf ‘‘measures’’ how sensitivefðaÞreacts to changes of the value of a atx. The estimate (SG) is also called ‘‘Efron-Stein inequality’’ and is an example of a concentration inequality. We refer to [24] for a review on concentration inequalities. Note that we can boundj@_xfðaÞjfrom above by appealing to the classical partial derivative:

j@xfðaÞj supffðaÞ fðaÞ~ : ~a2 with a¼a~ onZ^dn fxgg Z1

@fðaÞ

@aðxÞ

dx:

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

Lemma 4.4. Suppose thatPis independent and identically distributed, i.e.,

P¼ _x2Z^dP₀ðdxÞ for someP₀ probability measure on ₀: Thenhisatisfies (SG) with constant¼1.

Proof of Lemma 4.4. The argument is standard. We follow [15] and start with preparatory remarks.

. Letx₁;x₂;x₃;. . . denote an enumeration ofZ^d, . SincePis a product measure, we have

hjFx_ni ¼ Z

0

P₀ðdxnÞ:

. We introduce the shorthand

hi_n:¼ Z

ð₀Þⁿ

Yⁿ

i¼1

P₀ðdxiÞ;

n:¼ hi_n; ₀:¼; i.e.,n does not depend on the values ofaðx1Þ;. . .;aðxnÞ.

Thanks to the product structure ofP, we have

hj@_xnj²i ¼ hhj@_xnj²i_n1i ¼ Z

₀

P₀ðdx_nÞ

* 2+

n1

* +

Jensen

Z

0

P₀ðdxnÞ

n1

* 2+

¼ hjn1nj²i:

Now, the statement follows from the Martingale decomposition hð hiÞ²i ¼X¹

n¼1

hðn1nÞ²i: ð4:6Þ

Here comes the argument for ð4.6Þ: Since @_xhi ¼0, it suffices to consider 2L²ðÞ with hi ¼0. By a density argument, it suffices to consider2L²ðÞthat only depend on a finite number of coefficients, i.e.,_N ¼ hifor some N2N. Hence, by definition we have ₀¼ and_N¼ hi ¼0for N large enough, and thus (by telescopic sum)

¼X^N

n¼1

n1n: ð4:7Þ

Taking the square and the expected value yields h²i ¼X^N

n¼1

X^N

m¼1

hðn1nÞðm1mÞi:

Hence,ð4.6Þfollows, provided that the random variablesfn1ng_n2Nare independent (i.e., pairwise orthogonal in L²ðÞ). For the argument letm>n. Since by constructionm1mdoes not depend onaðy1Þ;. . .;aðym1Þwe have

_m1m¼ h_m1mi_m1; ð4:8Þ

and sincem1n, we have

hn1ni_m1¼ hhi_n1i_m1 hhi_ni_m1¼ hi_m1 hi_m1¼0: ð4:9Þ Hence, using the general identityhhui_m1vi ¼ huhvi_m1i, get

hð_m1mÞð_n1nÞi ¼^(4.8)hh_m1mi_m1ð_n1nÞi

¼ hðm1mÞhn1ni_m1i

(4.9)¼ 0

and the claim follows.

We next illustrate that (SG) not only implies, but also quantifies ergodicity. For this reason let pðt;xÞdenote the Green’s function for the heat equation @tþ rr (i.e., the unique function in Cð½0;1Þ; ‘²ðZ^dÞÞ \C¹ð; ‘²ðZ^dÞÞ satisfying@tpþ rrp¼0onð0;1Þ Z^d and pð0;xÞ ¼ðxÞ). Note that pðt;xÞ(which is also referred to as the heat kernel of the simple random walk onZ^d) is non-negative, normalizedP

x2Z^dpðt;xÞ ¼1, and in particular, it satisfies the on-diagonal heat kernel estimate

X

x2Z^d

p²ðt;xÞ CðdÞðtþ1Þ^d2: With help of pðt;xÞwe might define a semigroupðPðtÞÞ_t0onL²ðÞby setting

PðtÞ:L²ðÞ !L²ðÞ; PðtÞ:¼X

x2Z^d

pðt;xÞða;xÞ;

whereða;xÞ:¼ð_xaÞdenotes the stationary extension. Thanks to the interplay ofðÞandr, stationarity ofPimplies that the generator of ðP_tÞt0 is given by DD, whereDdenotes the horizontal derivative, see ð4.2Þ. We thus may equivalently write P_t¼expðtDDÞ. Note that the exponential is unambiguously defined, since DD:L²ðÞ ! L²ðÞis a bounded linear operator by the definition ofDDand the triangle inequality. It turns out that ergodicity can be characterized with help of Pt.

Lemma 4.5(Characterization and quantification of ergodicity). LetPbe stationary. Consider the semigroup defined by

PðtÞ:¼expðtDDÞ:

Then

(a) Pis ergodic, if and only if

82L²ðÞ: lim

t!1hjPðtÞ hij²i ¼0:

(b) IfPsatisfies (SG) with constant >0, then

hjPðtÞ hij²i¹² CðdÞ ffiffiffi

p ðtþ1Þ^d4X

x2Z^d

hj@xj²i

1 2;

Proof of Lemma 4.5 (a). We follow the argument in [15], and consider the space of shift-invariant functions IðÞ:¼ f2L²ðÞ:D¼0g;

and note that by definition,Pis ergodic, if and only ifIðÞ ¼R. (Indeed, this can be seen by considering first indicator functions of measurable sets, and then appealing to the fact that the linear span of indicator functions is dense inL²ðÞ).

Step 1. Claim:

IðÞ ¼ f2L²ðÞ:DD¼0g ¼kernel of DD:

The inclusion is trivial. Let2L²ðÞsatisfyDD¼0. Then 0¼ hDDi ¼ hjDj²i;

and thus2IðÞ.

Step 2. Claim:

IðÞ^?¼ fDF:F2L²ðÞ^dg (inL²ðÞ):

Since IðÞis closed, it suffices to prove

ðaÞ X:¼ fDF:F2L²ðÞ^dg IðÞ^? and ðbÞ X^?IðÞ:

Argument for (a):

8F2L²ðÞ^d; 2IðÞ: 0¼ hFDi ¼ hðDFÞi:

Argument for (b): Let2X^?. Then

8F2L²ðÞ^d : 0¼ hDFi ¼ hDFi ) 2IðÞ:

Step 3. (A priori estimates).

Let2L²ðÞand set uðtÞ:¼PðtÞ. Claim:

8t0 :hjuðtÞj²i h²i; ð4:10Þ

limt"1hjDuðtÞj²i ¼0: ð4:11Þ

Recall that@tuþDDu¼0 anduð0Þ ¼. Testing withuðtÞandDuðtÞyields 1

2 d

dthuðtÞ²i ¼ d

dtuðtÞuðtÞ

¼ hjDuðtÞj²i 0;

1 2

d

dthjDuðtÞj²i ¼ d

dtDuðtÞ DuðtÞ

¼ d

dtuðtÞDDuðtÞ

¼ hjDDuðtÞj²i 0:

Integration of the first identity yieldsð4.10ÞandR1

0 hjDuðtÞj²idt h²i<1, and thusð4.11Þ, sincet7!hjDuðtÞj²iis monotone (non-increasing) by the second estimate.

Step 4. (Conclusion).

Let2L²ðÞand write ¼⁰þ⁰⁰with⁰2IðÞ^? and⁰⁰2IðÞ. We claim that

PðtÞ!⁰⁰ inL²ðÞas t! 1: ð4:12Þ

Withð4.12Þwe can conclude the proof: IfPis ergodic, thenIðÞ ¼Rand⁰⁰¼ hi. On the other hand, ifPðtÞ! hi, then⁰⁰¼ hi. Since this is true for any and⁰⁰is the projection ontoIðÞ, we getIðÞ ¼R.

Argument for ð4.12Þ: Since IðÞ is the kernel of DD, we have PðtÞ¼PðtÞ⁰þ⁰⁰, and it suffices to prove PðtÞ⁰!0for all⁰2IðÞ^?. By Step 2 for any >0 we can findF2L²ðÞ^d withhj⁰DFj²i .ð4.10Þyields

8t2R_þ hjPðtÞð⁰DFÞj²i : ð4:13Þ

We claim that

lim

t"1hjPðtÞDFj²i ¼0: ð4:14Þ

Estimateð4.14Þcan be seen as follows. Since the shift operatorse₁;. . .; e_d commute, we get PðtÞDF¼expðtDDÞDF¼X^d

i¼1

D_iexpðtDDÞFi: Hence,

hjPðtÞDFj²i d X^d

i¼1

hjD_iPðtÞFij²i

stationarity¼ d X^d

i¼1

hjDiPðtÞFij²i:

Now,ð4.11Þimpliesð4.14Þ. Since >0arbitrary, the conclusion follows.

Proof of Lemma 4.5 (b). We follow the argument in [15].

Step 1. W.l.o.g. assume thathi ¼0. SetuðtÞ:¼PðtÞ, and recall that uðtÞ ¼X

z2Z^d

Gðt;zÞðzÞ:

We have

huðtÞi ¼X

z2Z^d

Gðt;zÞhðzÞi ¼0:

Thus, we can apply (SG) and obtain

hu²ðtÞi 1

X

y2Z^d

hj@_yuðtÞj²i: ð4:15Þ

We have

@yuðtÞ ¼X

z2Z^d

Gðt;zÞ@yuðt;zÞ

¼X

z2Z^d

Gðt;zÞ@yzuðt;zÞ:

The combination of both yields

X

y2Z^d

X

z2Z^d

Gðt;zÞ@yzuðt;zÞ 0

@

1 A

* 2+

0

@

1 A

1 2

¼ X

y2Z^d

X

x2Z^d

Gðt;yxÞ@xðyxÞ

!2

* +

0

@

1 A

1 2

4-inequality inðP

y2ZdhðÞ²iÞ¹2 X

x2Z^d

X

y2Z^d

hðGðt;yxÞ@xðyxÞÞ²i 0

@

1 A

1 2

¼

Gis deterministic;

stationarity X

x2Z^d

X

y2Z^d

G²ðt;yxÞhj@_xj²i 0

@

1 A

1 2

¼X

x2Z^d

hj@_xj²i¹² X

y2Z^d

G²ðt;yxÞ 0

@

1 A

1 2

:

We conclude by appealing to the on-diagonal heat kernel estimate X

y

G²ðt;yÞ ¼Gð2t;0Þ CðdÞðtþ1Þ^d2:

The estimate in part (b) of Lemma 4.5 extends to the semigroupexpðDðað0ÞDÞÞ. The extension is non-trivial, since on the one hand, the operatorrðað0ÞrÞand@x do not commute, and secondly, the regularity forrðað0ÞrÞis more involved than that for the discrete Laplacianrr. In [13] we obtained the following decay estimate:

Theorem 4.6 (see [13]). LetPbe stationary and satisfy (SG) with constant >0. Consider the semigroup given by PðtÞ:¼expðtDðað0ÞDÞÞ

Then for all exponents pwith p0ðd; Þ p<1, allt0 andF2L^2pðÞ^d we have hjPðtÞDFj^2pi

1

2p Cðd; ; ;pÞðtþ1Þ^ðd4þ12ÞX

y2Z^d

hj@_yFj^2pi

1 2p:

The proof of this theorem is out of the scope of this lecture. We only give some remarks: The exponent^d₄þ¹₂is optimal and the improvement of¹₂ compared to the exponent in Lemma 4.5 is due to the fact that in Theorem 4.6 we consider initial values in divergence form. The connection to homogenization is as follows: SetFðaÞ:¼ að0Þei and note that

X

y2Z^d

hj@_yFj^2pi

1

2p ¼ hj@₀Fj^2pi

1

2p sup

a;a⁰2

ja_iið0Þ a⁰_iið0Þj 1: Hence,hjPðtÞDFj^2pi

1

2p .ðtþ1Þ^ðd4þ12Þ. For d>2,ðtþ1Þ^ðd4þ12Þ is integrable onR_þ, and thus iðaÞ:¼

Z1 0

PðtÞDF dt2L^2pðÞ;

is well-defined and solves

Dað0ÞD_i¼DF; i.e.,Dðað0ÞðD_iþe_iÞÞ ¼0:

Now it is easy to see that the stationary extension_iða;xÞ:¼iðxaÞis a stationary solution to the corrector equation rðaðr_iþeiÞÞ ¼0 inZ^d,P-a.s.;

withhj_ij^2pi^2p1 Cðd; ; Þ. We can also consider the function defined ford2andT1by _T¼

Z1 0

exp t T

PðtÞDF dt:

From Theorem (4.6) we then deduce that

hj_Tj^2pi

1

2p Cðd; ; ;pÞ

log¹²T d¼2;p¼1, logT d¼2;p>1

1 d3.

8>

<

>:

By applyingDðað0ÞDÞtoT, we find that_T¹TþDðað0ÞðDTþÞÞ ¼0, and thus the stationary extension ofT 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 fora_homin the case of independent and identically distributed coefficients. In the following section we take a slightly different approach to obtain moment bounds which does not invoke the semigroup Pt.

4.3 Quantification of sublinearity in dimensiond2

In this section we prove (under a strong quantitative ergodicity assumption) that (high) moments ofrandr are bounded, and we quantify the growth rate ofhjðxÞj²iandhj ðxÞj²i. The argument that we present combines the strategy of [5] (which relies on a Logarithmic Sobolev inequality to quantify ergodicity) and ideas of [12], where optimal growth rates for the correctors are obtained in the continuum setting and for strongly correlated coefficients. We also refer to [4, 18] where similar estimate (that are stronger in terms of stochastic integrability) are obtained for coefficients satisfying a finite range of dependence condition (instead of the concentration inequality that we assume). Except for some input from elliptic regularity theory (that we detail below), the argument that we present is self-contained. We start by introducing our quantitative ergodicity assumption onP. Instead of the absolute value of the vertical derivative

@_xf, see Definition 4.3, we appeal to the ‘‘Lipschitz derivative’’

j@lip;xfðaÞj:¼supfjfða⁰Þ fða⁰⁰Þj:a⁰;a⁰⁰2;a¼a⁰¼a⁰⁰inZ^dn fxgg:

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

f²log f² hf²i

1 2

X

x2Z^d

hj@lip;xfj²i:

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

f ¼1þ"f⁰in 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 stochastic homogenization.

Our main result is the following:

Theorem 4.8. SupposePsatisfies (D1) and (LSI) with constant >0. Letði; iÞdenote the extended corrector of Proposition 4.1. Then for all p1we have

hjrj^2pþ jr j^2pi

1

2p Cðp; ;d; Þ and for allx2Z^d we have

hjðxÞj^2pþ j ðxÞj^2pi

1

2p Cðp; ;d; Þ log¹²ðjxj þ2Þ d¼2,

1 d3.

(

Note that the estimate is uniformxford3. In that case we can find stationary extended correctors, i.e.,ð; Þsatisfy ð; Þða;xþyÞ ¼ ð; Þðxa;yÞ instead of the anchoring condition ð; Þð0Þ ¼0. In dimension d¼2 the correctors diverge logarithmically. The logarithm (and the exponent¹₂) 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 the remainder Z of the two-scale expansion satisfies the estimate, for all p1,

X

Z^d

jZj²þjrZj²

!p

* +_2p¹

. X

x2Z^d

jru0ðxÞj²!dðxÞ þX

x2Z^d

jrru0ðxÞj²!dðxÞ

!¹₂

; where

!dðxÞ:¼ logðjxj þ2Þ d¼2,

1 d3,

and.meansup to a constant that only depends ond; ; andp. Ford3standard‘²-regularity shows that the right-hand side is bounded by kfk_‘2. Likewise, for d¼2, weighted ‘²-regularity shows that the right-hand side is estimated bykf ffiffiffiffiffiffi!d

p k_‘2. Overall we obtain the estimate X

Z^d

jZj²þjrZj²

!p

* +_2p¹

. X

Z^d

jfj²!d

!¹₂ :

For a comparison with Theorem 3.3 we need to pass to the scaled quantities Z_":"Z^d!R, Z_"ðxÞ:¼Zð^x_"Þ, ri;"Z_"ðxÞ:¼"¹ðriZÞð^x_"Þ, and f_"ðxÞ:¼"²fð^x_"Þ. The previous estimate than turns into

X

"Z^d

jZ_"j²þjrZ_"j²

!p

* +_2p¹

. X

Z^d

jf_"j²!_d

!¹₂

"log¹²ð¹_"þ2Þ d¼2,

" d3.

(

Thus, ford¼2 we obtain a different scaling in".

A continuum version of Theorem 4.8 (with optimal stochastic integrability) has been recently obtained in [12]. In the discrete case the result ford3is a corollary of Theorem 4.6, while ford¼2the 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 the following 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 onDe Giorgi-Nash-Moser theory, see Lemma 4.11;

(b) aweighted Meyer’s estimateestablished in [5], see Lemma 4.12 below;

(c) anannealed Green’s function estimatefor high moments ofjrxryGðx;yÞjestablished 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 alla2. Estimate (c), which invokes the expectation, has a different nature and is a first example of a large scale regularity result for elliptic operator with stationary and ergodic coefficients. We refer to the recent work [11] where a rather complete large scale regularity theory is developed. For a another approach to large scale regularity that is based on linear mixing conditions 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 a2 the Green’s function (which is non-negative) satisfies

Gða;x;yÞ Cðd; Þ logðjxj þ2Þ d¼2, ðjxj þ1Þ^2d d>2.

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

Lemma 4.12(weighted Meyer’s estimate, see Proposition 1 in [5]). There existsq0>1and0>0(only depending ond and) such that for anya2and anyv:Z^d !Rand h:Z^d!R^d related by

rðarvÞ ¼ rrh inZ^d; the following estimates hold:

(a) For allðq; Þ 2 ½1;q₀ ½0; ₀we have X

x2Z^d

jrvðxÞj^2qðjxj þ1ÞCðd;q; ÞX

x2Z^d

jrhðxÞj^2qðjxj þ1Þ: ð4:16Þ (b) For1<qq01 andL2 consider the weight

!_q;LðxÞ:¼ ðjxj þ1Þ^2ðq1ÞþL^2ð1qÞðjxj þ1Þ^4ðq1Þ d ¼2,

ðjxj þ1Þ^2dðq1Þ d 3.

Then we have

X

x2Z^d

jrvðxÞj^2q!_q;LðxÞ Cðd;qÞX

x2Z^d

jrhðxÞj^2q!_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–Zygmund estimate forrr, see Proposition 1 in [5]. In the continuum case the estimates are classical. Note that the weight in ð4.17Þsatisfies

X

x2Z^d

!

1 q1

q;L ðxÞ

!

¼Cðd;qÞ logL d¼2,

1 d3. 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 existsq0>1and0>0(only depending ondand) such that for allðq; Þ 2 ½1;q0 ½0; 0we have

sup

a2

X

x2Z^d

jrrGða;x;0Þj^2qðjxj þ1ÞCðq; ;d; Þ: ð4:19Þ

Proof. Note that we have

r_xðar_xry;iGða;;yÞÞ ¼ ðr_iÞð yÞ:

Hence, the estimate follows fromð4.16Þ.

Lemma 4.14(annealed Green’s function estimate, see [25]). SupposePsatisfies (D1) and (LSI). Then for allp1 we have

hjrxryGða;x;yÞj^2pi

1

2p Cðd; ; Þðjxyj þ1Þ^d: For a proof see [25].

Sensitivity estimate and proof of Theorem 4.8

Lemma 4.15(Sensitivity estimate). SupposePsatisfies (D1) and (D2). Then there exists⁰withPð⁰Þ ¼1such that for i¼1;. . .;d, alla2⁰and all x2Z^d we have

j@lip;xriða;yÞj Cðd; ÞjrrGða;y;xÞjjriða;xÞ þeij:

Proof of Lemma 4.15. We define ⁰ as the set of all a2 such that equations ð4.3Þ and ð4.5Þ admit for i;j;k¼1;. . .;d, sublinearly growing (and thus unique) solutions with iða;0Þ ¼0 and ijkða;0Þ ¼0. By Proposition 4.1 we havePð⁰Þ ¼1. Furthermore, note thatð4.5Þcan rewritten as

rr ijk¼ rQijk; where

Qijkða;xÞ:¼ ðqiða;xþejÞ ekÞej ðqiða;xþekÞ ejÞek: ð4:20Þ (Indeed, this follows from the identityriuðxÞ ¼ ðr_iuÞðxþe_iÞ).

Step 1. Leta2⁰ anda⁰2⁰witha¼a⁰ inZ^dn fxg. Seta¼aa⁰and

i:¼iða;Þ iða⁰;Þ; ijk:¼ ijkða;Þ ijkða⁰;Þ;

qi:¼qiða;Þ qiða⁰;Þ; Qijk:¼Qijkða;Þ Qijkða⁰;Þ:

Then a direct calculation (using ð4.3Þ–ð4.5Þ, and the fact thataðyÞ ¼0 for ally6¼x) yields

rðariÞ ¼ rðaðriða⁰;Þ þeiÞÞ; ð4:21Þ

rr _ijk¼ rQ_ijk; ð4:22Þ

qiðyÞ ¼aðyÞðrða;xÞ þeiÞ þa⁰ðyÞriðyÞ ð4:23Þ Q_ijk¼ ðaðyÞðrða;xþe_jÞ þe_iÞ e_kÞe_j ð4:24Þ

ðaðyÞðrða;xþekÞ þeiÞ ejÞek

þ ða⁰ðyÞðriðyþejÞ ekÞej

ða⁰ðyÞr_iðyþe_kÞ e_jÞe_k:

Sinceiand ijkare sublinear (as differences of sublinear functions), we may test with the Green’s function and get riðyÞ ¼ ryrxGða;y;xÞ aðxÞðriða⁰;xÞ þeiÞ ð4:25Þ Applyingð4.25Þwithy¼xand the roles ofaanda⁰ interchanged, yields

ðrða⁰;xÞ þeiÞ ðrða;xÞ þeiÞ ¼ rxrxGða⁰;x;xÞðriða;xÞ þeiÞ;

and thus

jrða⁰;xÞ þeij ðjrxrxGða⁰;x;xÞj þ1Þjriða;xÞ þeij ð4:26Þ ð¹þ1Þjriða;xÞ þeij:

We conclude that

jr_iðyÞj ¼Cðd; ÞjryrxGða;y;xÞjjr_iða;xÞ þe_ij ð4:27Þ Step 2.

We claim thata2⁰,a⁰2witha¼a⁰onZ^dn fxgimplies thata⁰2⁰. In view of the definition of⁰, we need to show existence of sublinear solutions toð4.3Þandð4.5Þfora⁰. Indeed, this can be inferred as follows: Equationsð4.21Þ andð4.22Þadmit unique sublinear solutions_iand _ijkwith_ið0Þ ¼ _ijkð0Þ ¼0, since the right-hand side ofð4.21Þ is the divergence of a compactly supported function, and the right-hand side ofð4.22Þis the divergence of a square summable function. Now the sought for sublinear solutions are given by _iða⁰;Þ:¼_iða;Þ þ_i and _ijkða⁰;Þ ¼

ijkða;Þ þ ijk. As a consequence of this stability of⁰w.r.t. compactly supported variations ofa, when estimating j@lip;xfðaÞjfora2⁰, we only need to take the sup (in the definition of the Lipschitz derivative) over fieldsa⁰;a⁰⁰2⁰ witha¼a⁰¼a⁰⁰inZ^dn fxginto account. Thus, the claimed estimate follow fromð4.27Þ.

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

Lemma 4.16. LetPsatisfy (LSI) with constant >0. Then for any1 p<1, any >0and all random variables f we have the estimates

hjf hfij^2pi

1

2p Cðp; Þ X

x2Z^d

j@lip;xfj²

!p

* +_2p¹

; ð4:28Þ

hjfj^2pi

1

2p Cð;p; Þhjfj²i¹²þ X

x2Z^d

j@lip;xfj²

!p

* +_2p¹

: ð4:29Þ

Estimateð4.28Þfor p¼1is the usual Spectral Gap estimate, which is implied by (LSI).ð4.28Þfor p>1follows from the estimate for p¼1 by the argument in [13]. For a proof ofð4.29Þwe refer to [25]. We are now in position to establish moment bounds for ri andr i:

Lemma 4.17. SupposePsatisfies (D1) and (LSI). Then for all1p<1 hjriþeij^2pþ jr ij^2pi

1

2p Cðp; ;d; Þ:

Proof. Step 1. Proof of the bound forri.

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