# Sublinear growth of the corrector in stochastic homogenization: optimal stochastic estimates for slowly decaying correlations

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## Abstract

We establish sublinear growth of correctors in the context of stochastic homogenization of linear elliptic PDEs. In case of weak decorrelation and “essentially Gaussian” coefficient fields, we obtain optimal (stretched exponential) stochastic moments for the minimal radius above which the corrector is sublinear. Our estimates also capture the quantitative sublinearity of the corrector (caused by the quantitative decorrelation on larger scales) correctly. The result is based on estimates on the Malliavin derivative for certain functionals which are basically averages of the gradient of the corrector, on concentration of measure, and on a mean value property for *a*-harmonic functions.

## Keywords

Stochastic homogenization Linear elliptic equation Homogenization corrector Sow decorrelation## Mathematics Subject Classification

35B27 35J15 35R60## 1 Introduction

*a*is typically a uniformly elliptic and bounded coefficient field, chosen at random according to some stationary and ergodic ensemble \(\langle \cdot \rangle \). On large scales (and for slowly varying

*f*), one may then approximate the solution

*u*to the Eq. (1) by the solution \(u_{hom}\) to the so-called

*effective equation*

*corrector*(cf. below for a definition of the corrector).

The goal of the present paper is to provide a fairly simple proof of *quantified* sublinear growth of the corrector under very mild assumptions on the decorrelation of the coefficient field *a* under the ensemble \(\langle \cdot \rangle \). We do this in the context of coefficient fields that are essentially Gaussian. More precisely, we consider coefficient fields *a* which are obtained from a Gaussian random field by pointwise application of some (nonlinear) mapping, the role of the nonlinear map being basically to enforce uniform ellipticity and boundedness of our coefficient field.

The motivation for this result is the following: Gloria et al. [11] have shown that *qualitatively* sublinear growth of the (extended) corrector \((\phi ,\sigma )\) (cf. below for a definition) entails a large-scale intrinsic \(C^{1,\alpha }\) regularity theory for *a*-harmonic functions. In a subsequent work [10], the two authors of the present paper have shown that *slightly quantified* sublinear growth of the corrector even leads to a large-scale intrinsic \(C^{k,\alpha }\) regularity theory for any \(k\in {\mathbb {N}}\). Therefore, the results of the present work show that even in case of ensembles with very mild decorrelation, for almost every realization of the coefficient field, *a*-harmonic functions have arbitrary intrinsic smoothness properties on large scales. Furthermore, our results enable us to estimate the scale above which this happens—a random quantity—in a stochastically optimal way. Indeed, the motivation for the present work was to establish such a (necessarily intrinsic) higher order regularity theory under the weakest possible assumptions on the decay of correlations.

By an *intrinsic* regularity theory we mean that the regularity is measured in terms of objects intrinsic to the Riemannian geometry defined by the coefficient field *a*, like the dimension of the space of *a*-harmonic functions of a certain algebraic growth rate, or like estimates on the Hölder modulus of the derivative of *a*-harmonic functions as measured in terms of their distance to *a*-linear functions. An *extrinsic* large-scale regularity theory for *a*-harmonic functions in case of random coefficients was initiated on the level of a \(C^{0,\alpha }\) in [7, 21] and pushed to \(C^{1,0}\) in [4], which significantly extended qualitative arguments from the periodic case [5] to quantitative arguments in the random case. However, an extrinsic regularity theory is limited to \(C^{1,0}\), as can be seen considering the harmonic coordinates: Taking higher order polynomials into account does not increase the local approximation order.

After this motivation, we now return to the discussion of the history on bounds on the corrector, as they depend on assumptions on the stationary ensemble of coefficient fields. Almost-sure sublinearity (always meant in a spatially averaged sense) of the corrector \(\phi \) under the mere assumption of ergodicity was a key ingredient in the original work on stochastic homogenization by Kozlov [19] and by Papanicolaou and Varadhan [24]. Almost-sure sublinearity of the *extended* corrector \((\phi ,\sigma )\), as is needed for the large-scale intrinsic \(C^{1,\alpha }\)-regularity theory, was established in [11] under mere ergodicity.

Yurinskii [25] was the first to quantify sublinear growth under general mixing conditions, however only capturing suboptimal rates even in case of finite range of dependence. Very recently, a much improved quantification of sublinear growth of \(\phi \) under finite range assumptions was put forward by Armstrong et al. [1], relying on a variational approach to quantitative stochastic homogenization introduced by Armstrong and Smart [4], an approach which presumably can be extended to the case of non-symmetric coefficients and more general mixing conditions following [3]. Recently, optimal growth bounds on the corrector \(\phi \) with optimal—i.e. Gaussian—stochastic integrability have been established under the assumption of finite range of dependence [2, 17].

Optimal growth rates have been obtained under a quantification of ergodicity different from finite range or mixing conditions, namely under Spectral Gap assumptions on the ensemble. This functional analytic tool from statistical mechanics was introduced into the field of stochastic homogenization in an unpublished paper by Naddaf and Spencer [23], and further leveraged by Conlon et al. [8, 9], yielding optimal rates for some errors in stochastic homogenization in case of a small ellipticity contrast. The work of Gloria, Neukamm and the second author extended these results to the present case of arbitrary ellipticity contrast [12, 13, 14], in particular yielding at most logarithmic growth of the corrector (and its stationarity in \(d>2\)). Loosely speaking, the assumption of a Spectral Gap Inequality amounts to correlations with integrable tails; in the above-mentioned works it has been used for discrete media (i.e. random conductance models), but has subsequently been extended to the continuum case [15, 16].

A strengthening of the Spectral Gap Inequality is given by the Logarithmic Sobolev Inequality (LSI); it is a slight strengthening in terms of the assumption (still essentially encoding integrable tails of the correlations), but a substantial improvement in its effect, since it implies Gaussian concentration of measure for Lipschitz random variables. The assumption of LSI and implicitly concentration of measure, which will be explicitly used in this work, has been introduced into stochastic homogenization by Marahrens and Otto [21]. In [11], it has been shown that the concept of LSI can be adapted to also capture ensembles with slowly decaying correlations, i.e. thick non-integrable tails, by adapting the norm of the vertical or Malliavin derivative to the correlation structure. As a result, the stochastic integrability of the optimal rates could be improved from algebraic to (stretched) exponential, but missing the expected Gaussian integrability.

The main merit of the present contribution w.r.t. to [11] is twofold: First, our approach directly provides optimal quantitative sublinearity of the corrector \((\phi ,\sigma )\) on all scales above a random minimal radius \(r_*\), i.e. in contrast to the estimates of [11] our estimates capture the decorrelation on scales larger than \(r_*\) in a single argument. Note that our definition of \(r_*\) differs from the one in [11]. Second, in case of weak decorrelation, our simpler arguments are nevertheless sufficient to establish optimal stochastic moments for the minimal radius \(r_*\) above which the corrector \((\phi ,\sigma )\) displays the quantified sublinear growth.

*a*where

*a*is given by \(a(x):=\Phi ({\tilde{a}}(x))\). Note that the normalization in the constant in (3) and in the Lipschitz constant is not essential, since it can be achieved by a rescaling of

*x*and the amplitude of \({\tilde{a}}\).

Concerning the mathematical tools of our approach, several ideas are inspired by the work [11]. In particular, a key component of our approach are sensitivity estimates (Malliavin derivative bounds) for certain integral functionals, which basically average the gradient \(\nabla (\phi ,\sigma )\) over an appropriate cube. Furthermore, we rely on a mean-value property for *a*-harmonic functions, which has been derived in [11] under appropriate smallness assumptions on the corrector. In our present contribution, we however pursue a conceptually simpler route to estimate the Malliavin derivative: The sensitivity estimate is performed through appropriate \(L^q\)-norm bounds and Meyer’s estimate, rather than a more involved \(\ell ^2-L^1\)-norm bound like in [11].

*a*-harmonic map. In the context of stochastic homogenization, one is therefore interested in constructing random scalar fields \(\phi _i=\phi _i(a,x)\) subject to the equation

*x*: The \(\phi _i\) then facilitate the transition from the \(a_{hom}\)-harmonic (Euclidean) coordinates \(x\mapsto x_i\) to the “

*a*-harmonic coordinates” \(x\mapsto x_i+\phi _i(x)\). Since any affine map may be represented in the form \(b+\sum _i \xi _i x_i\) for \(b,\xi _i\in {\mathbb {R}}\), the \(\phi _i\) also facilitate the construction of associated

*a*-harmonic “corrected affine maps” \(b+\sum _i \xi _i(x_i+\phi _i)\).

*a*—note that in this picture, one has \(f\equiv 0\) in (1)—, the quantity \(E_i:=e_i+\nabla \phi _i\) corresponds to the (curl-free) “microscopic” electric field associated with a “macroscopic” electric field \(e_i\) (and, therefore, \(\phi _i\) corresponds to the “microscopic” correction to the “macroscopic” electric potential \(x_i\)). The corresponding (divergence-free) “microscopic” current density is given by

**Notation**To quantify the ellipticity and boundedness of our coefficient fields, throughout the paper we shall work with the assumptions

*a*does not induce a loss of generality of our results.

For our convenience, throughout the paper we shall assume that our coefficient field *a* is symmetric. The arguments however easily carry over to the case of non-symmetric coefficient fields by simultaneously considering the correctors for the dual equation (i.e. the PDE with coefficient field \(a^*\), \(a^*\) denoting the transpose of *a*).

The expression \(s\lesssim t\) is an abbreviation for \(s\le C t\) with *C* a generic constant only depending on the dimension *d*, the exponent \(\beta >0\), and the ellipticity ratio \(\lambda >0\).

The expression \(s\ll t\) stands for \(s\le \frac{1}{C} t\) with *C* a generic sufficiently large constant only depending on the dimension *d*, the exponent \(\beta >0\), and the ellipticity ratio \(\lambda >0\).

By *I*(*E*) we denote the characteristic function of an event *E*.

The notation Open image in new window refers to the average integral over the set *A*, i.e. we have Open image in new window .

In the sequel, \((\phi ,\sigma )\) stands for any component \(\phi _i,\sigma _{ijk}\) for \(i,j,k=1,\ldots ,d\).

## 2 Main results and structure of proof

Let us now state our main theorem. To quantify the sublinear growth of the extended corrector \((\phi ,\sigma )\), we first quantify the decay of spatial averages of \(\nabla (\phi ,\sigma )\) over larger scales. In view of the decorrelation assumption (3) for our ensemble of coefficient fields, we expect that, up to logarithms, it is the exponent \(\frac{\beta }{2}\) that governs the decay of averages of \(\nabla (\phi ,\sigma )\) and the improvement over linear growth for \((\phi ,\sigma )\). Indeed, this exponent is reflected in the theorem.

### Theorem 1

*a*, where

*a*is the image of \({\tilde{a}}\) under pointwise application of the map \(\Phi \), i.e. \(a(x):=\Phi ({\tilde{a}}(x))\).

*C*denotes a generic constant only depending on

*d*and \(\lambda \).

- (i)Consider a linear functional \(F=Fh\) on vector fields \(h=h(x)\) satisfying the boundedness propertyfor some radius \(r>0\). Then the random variable \(F\nabla (\phi ,\sigma )\) satisfies uniform Gaussian bounds in the sense of$$\begin{aligned} |Fh|\le \left( -\int _{|x|\le r}|h|^\frac{2d}{d+\beta }dx\right) ^\frac{d+\beta }{2d} \end{aligned}$$(14)$$\begin{aligned} \langle I(|F\nabla (\phi ,\sigma )|\ge M)\rangle \le C\exp \left( -\frac{1}{C}r^\beta M^2\right) \quad \hbox {for all}\;M\le 1. \end{aligned}$$(15)
- (ii)There exists a (random) radius \(r_*\) for which the “iterated logarithmic” boundholds and which satisfies the stretched exponential bound$$\begin{aligned} \frac{1}{r^2}-\int _{|x|\le r}|(\phi ,\sigma )--\int _{|x|\le r}(\phi ,\sigma )|^2dx\le \left( \frac{r_*}{r}\right) ^\beta \log \left( e+\log \left( \frac{r}{r_*}\right) \right) \quad \hbox {for}\;r\ge r_*\nonumber \\ \end{aligned}$$(16)$$\begin{aligned} \left\langle \exp \left( \frac{1}{C}r_*^\beta \right) \right\rangle \le C. \end{aligned}$$(17)

Morally speaking, Theorem 1 converts statistical information on the coefficient field *a* (or rather \({\tilde{a}}\)) into statistical information on the coefficient field \(\nabla \phi :=\nabla (\phi _1,\ldots ,\phi _d)\) related by (4). Despite the nonlinearity of the map \(a\mapsto \nabla \phi \), which only in its linearization around \(a=\mathrm{id}\) turns into the Helmholtz projection, Theorem 1 states that \(\nabla \phi \) essentially inherits the statistics of *a*: (15) implies in particular that spatial averages \(F=-\int _{|x|\le r}\nabla \phi dx\) of \(\nabla \phi \) satisfy the same bounds as if \(\nabla \phi \) itself was Gaussian with correlation decay (3). On the level of these Gaussian bounds, the only price to pay for the nonlinearity is the restriction \(M\lesssim 1\) in (15) on the threshold.

Incidentally, the way we obtain (ii) from (i) bears similarities with an argument in [1] in the sense that a decomposition into Haar wavelets is implicitly used.

*i*,

*j*,

*k*nonzero) constant \(C(d,\beta ,i,j,k)\). The difference Open image in new window is therefore a centered Gaussian random variable with variance \(\sim r^{2-\beta }\), which entails that the moment bound (17) for the factor \(r_*^\beta \log (e+\log (r/r_*))\) in the estimate (16) is (almost) optimal. The scaling in

*r*of the bound (16) is optimal in view of the law of the iterated logarithm; details are provided in the “Appendix”.

To obtain an estimate like (15), the starting point of our proof is the Gaussian concentration of measure applied to \(\tilde{a}\). Recall the notion of the covariance operator \({\text {Cov}}\), which in our setting of a stationary centered Gaussian random field \({\tilde{a}}\) is given as the convolution with the tensor field \(\langle {\tilde{a}}(x) \otimes {\tilde{a}}(0) \rangle \).

### Proposition 1

*F*, that is, a function(al) \(F=F({\tilde{a}})\). Suppose that

*F*is 1-Lipschitz in the sense that its functional derivative, or rather its Fréchet derivative with respect to \(L^2({\mathbb {R}}^d;{\mathbb {R}}^{d\times d})\), \(\frac{\partial F}{\partial {\tilde{a}}}=\frac{\partial F}{\partial {\tilde{a}}}({\tilde{a}},x)\), which can be considered a random tensor field and assimilated with a Malliavin derivative, satisfies

*F*has Gaussian moments in the sense of

We now substitute our assumption (21) on the Fréchet derivative by a stronger but more tractable condition.

### Lemma 1

*F*on the space of tensor fields \({\tilde{a}}\) of the form \(F=F(a)\) with \(a(x):=\Phi ({\tilde{a}}(x))\); we shall use the abbreviation \(F(\tilde{a})\) for \(F(\Phi ({\tilde{a}}))\). Let \(q\in (1,2)\) be given by

*F*with respect to \(L^2({\mathbb {R}}^d;{\mathbb {R}}^{d\times d})\) satisfies

We observe that if *q* and \(\beta \) are related by (24), as \(\beta \uparrow d\) we have \(q\uparrow 2\) and for \(\beta \downarrow 0\) we have \(q\downarrow 1\).

For linear functionals of (the gradient of) the corrector (which are therefore nonlinear functionals of the coefficient field *a*), we now establish an explicit representation of the Fréchet derivative; this will aid us in verifying the Lipschitz condition (25) and thus ultimately the concentration of measure statements (22) and (23) for (an appropriate modification of) such functionals.

### Lemma 2

*a*be some coefficient field subject to the ellipticity and boundedness conditions (10), (11). Then the following two assertions hold:

- (1)Consider the Fréchet derivative \(\frac{\partial F}{\partial a}\) of the functional \(F:=F\nabla \sigma _{ijk}\) (note that this functional is nonlinear in
*a*, although it is linear in \(\sigma _{ijk}\)) at*a*(for some fixed*i*,*j*,*k*). Introduce the decaying solutions*v*, \({\tilde{v}}_{jk}\) to the equationsand (where \(a^*\) denotes the transpose of$$\begin{aligned} -\triangle v=\nabla \cdot g \end{aligned}$$(27)*a*)We then have the representation$$\begin{aligned} -\nabla \cdot a^*\left( \nabla {\tilde{v}}_{jk}+(\partial _j ve_k-\partial _k ve_j)\right) =0. \end{aligned}$$(28)$$\begin{aligned} \frac{\partial F}{\partial a}(a)=\left( \partial _jve_k-\partial _kve_j+\nabla \tilde{v}_{jk}\right) \otimes (\nabla \phi _i+e_i). \end{aligned}$$(29) - (2)Consider the Fréchet derivative \(\frac{\partial F}{\partial a}\) of the functional \(F:=F\nabla \phi _i\) at
*a*. Introduce the decaying solution \({\overline{v}}\) to the equation (again, \(a^*\) denoting the transpose of*a*)We then have the representation$$\begin{aligned} -\nabla \cdot a^*\nabla {\overline{v}} = \nabla \cdot g. \end{aligned}$$(30)$$\begin{aligned} \frac{\partial F}{\partial a}(a) = \nabla {\overline{v}} \otimes (\nabla \phi _i + e_i). \end{aligned}$$(31)

The previous explicit representation of the Fréchet derivative for certain linear functionals of (the gradient of) the corrector \((\phi ,\sigma )\) enables us to verify the bound (25) for the Malliavin derivative, provided that a certain mean value property is satisfied for *a*-harmonic functions. Note that the latter requirement is a condition on the coefficient field *a*; in Lemma 4 below we shall provide a sufficient condition for this property to hold.

As the functionals which the next lemma shall be applied to are basically averages of \(\nabla \phi \) or \(\nabla \sigma \) over cubes of a certain scale *r*, we state the lemma in a form which makes it directly applicable in such a setting. In particular, the boundedness assumption (32) for the linear functional is motivated by these considerations.

### Lemma 3

*p*through

*a*) at some symmetric coefficient field

*a*subject to the conditions (10), (11).

*a*is such that the mean value propertyholds for any

*a*-harmonic function

*u*and provided that furthermore

*a*is such thatis satisfied, we have the estimate

Note that for *q* related to \(\beta \) through (24) and *p* related to *q* through (34), we have \(r^{-\frac{(p-2)d}{p}}=r^{-\beta }\), i.e. by (37) the \(L^q\)-norm of the Malliavin derivative decays like \(r^{-\frac{\beta }{2}}\). This demonstrates that for functionals like our averages of \(\nabla (\phi ,\sigma )\)—note that these functionals have vanishing expectation due to the vanishing expectation of \(\nabla (\phi ,\sigma )\)—, the concentration of measure indeed improves on large scales with the desired exponent: The “typical value” of the average of \(\nabla (\phi ,\sigma )\) on some scale *r* decays like \(r^{-\frac{\beta }{2}}\).

We now have to provide a sufficient condition for the mean value property for *a*-harmonic functions (35). To do so, we make use of the following result from [11], which provides the mean-value property assuming just an appropriate sublinearity condition on the corrector \((\phi ,\sigma )\).

### Proposition 2

*d*and ellipticity ratio \(\lambda >0\) with the following property: Suppose that for an elliptic coefficient field

*a*subject to the ellipticity and boundedness conditions (10) and (11) the scalar and vector potentials \((\phi ,\sigma )\), cf. (4) and (7), satisfy

*a*-harmonic function

*u*in \(\{|x|\le R\}\) we have

We shall show in the proof of the next lemma that the quantitative sublinearity condition on the corrector (38) may be reduced to a smallness assumption on a certain family of linear functionals of the gradient of the corrector. This reduction relies on the compactness of the left-hand side of (38) with respect to the \(L^2\)-norm of \(\nabla (\phi ,\sigma )\), which in turn may be estimated via Caccioppoli’s estimate by the left-hand side. It appeals to a quantitative version of inequalities in functional analysis where an intermediate norm is estimated by a bit of a stronger norm and a lot of a weaker (semi-)norm, the role of which is played by the expression in (39). A slight subtlety follows from the fact that the use of Caccioppoli’s inequality increases the radius (by a factor of two, say), so that one has to buckle on the level of all dyadic radii *R* larger than the given radius *r*, cf. the expression in (39). This requires the qualitative a priori information (36). One has a lot of flexibility in the choice of the functionals \(F_n\); for pure convenience we choose the same functionals, of Haar wavelet-type, that play a prominent role in the proof of Assertion (ii) of Theorem 1. Other natural choices would be the first *N* eigenfunctions of the Neumann-Laplacian, like in Step 7 of the proof of Theorem 2 in [6] or the proof of Lemma 2.6 in [15].

### Lemma 4

*a*-harmonic function

*u*on scales \(\ge r\), i.e. we have

*a*-harmonic functions on scales larger than

*r*. For the concentration of measure estimate (22), however, an unconditional estimate of the form (21) or (25) (the latter being a proxy for (21)) is needed. By Lemma 4 we know that the mean-value property holds, provided that for a certain family of linear functionals of the corrector the smallness estimate

The proof of the second assertion of our main theorem will mainly rely on the first assertion of the theorem as well as the quantitative improvement of the Malliavin derivative of averages of \((\nabla \phi ,\nabla \sigma )\) on larger scales, as captured by the estimate (37).

## 3 Concentration of measure and estimates of the Malliavin derivative

### 3.1 Concentration of measure

### Proof of Proposition 1

For the proof of the concentration of measure estimate (22), we refer the reader to [20, Proposition 2.18]. We now establish (23). By Chebychev’s inequality, (22) implies \(\langle I(F-\langle F\rangle \ge M)\rangle \le \exp (-\frac{M^2}{2})\). In combination with the same estimate with *F* replaced by \(-F\), we obtain (23). \(\square \)

### Proof of Lemma 1

We need to verify that the condition (21) is implied by the assumption (25).

*q*and \(\beta \) are related by (24). From this string of inequalities we learn that (21) also holds provided

### 3.2 Representation of the Malliavin derivative

### Proof of Lemma 2

We first give the argument for the “vector potential” \(\sigma \), fixing a component \(\sigma _{ijk}\). Consider a functional of the form \(F:=F\nabla \sigma _{ijk}\) with *Fh* as in (26). We claim that the Fréchet derivative of *F* with respect to *a* is given by (29) where the functions \(v=v(x)\) and \(\tilde{v}_{jk}={\tilde{v}}_{jk}(a,x)\) are determined as the decaying solutions of the elliptic Eqs. (27) and (28).

*F*as a function of

*a*amounts to a linearization. We thus consider an arbitrary tensor field \(\delta a=\delta a(x)\), which we think of as an infinitesimal perturbation of

*a*, and which thus generates infinitesimal perturbations \(\delta \phi \) and \(\delta \sigma \) of \(\phi \) and \(\sigma \) according to (4), (6), and (9), that is,

*F*, this implies by integration by parts (or rather by directly appealing to the weak Lax–Milgram formulations of the elliptic equations)

*F*with respect to

*a*is given by

### 3.3 Sensitivity estimate

### Proof of Lemma 3

We now argue that under certain boundedness assumptions on \(F=Fh\) as a linear functional in vector fields \(h=h(x)\), we control the size (25) of its Fréchet derivative \(\frac{\partial F}{\partial a}=\frac{\partial F}{\partial a}(a,x)\) as a nonlinear functional \(F\nabla \sigma _{ijk}=F(a)\) in coefficient fields \(a=a(x)\) (and similarly in the case \(F(a)=F\nabla \phi _i\); for this case, the (simpler) proof is sketched afterwards).

*p*and its dual exponent \(\frac{p}{p-1}\): For any decaying function

*w*and vector field

*h*on \({\mathbb {R}}^d\) related by

*a*as well as the estimate \(|p-2|\ll 1\), which is ensured by our condition (33). Note that an analogous estimate would hold for the dual equation \(-\nabla \cdot a^*\nabla w=\nabla \cdot h\) if our coefficient field were nonsymmetric.

*B*; we write \(\Vert \cdot \Vert _{p}\) when \(B={\mathbb {R}}^d\). We start by arguing that because \(\frac{p}{p-1}\in (1,2)\), (35) also entails

*m*is the spatial average of

*u*on \(\{|x|\le R\}\). The combination of the last two estimates yields

*a*. Using this representation, a partition into dyadic annuli, and Hölder’s estimate (recall (34)) we obtain

*a*-harmonic function \(u(x)=x_i+\phi _i(x)\), cf. (4), we obtain for all radii \(\rho \ge r\) using Caccioppoli’s inequality and (36)Hence (50) turns into

*v*and \(\tilde{v}_{jk}\). The estimate of the terms in line (51) is easy: By (48) and Calderon–Zygmund for (27) we obtain \(\Vert \nabla v\Vert _{p}\lesssim \Vert g\Vert _{p}\le r^{-\frac{p-1}{p}d}\). By (44) we have Calderon-Zygmund with exponent

*p*for the equation (28), so that \(\Vert \nabla {\tilde{v}}_{jk}\Vert _{p}\lesssim \Vert \nabla v\Vert _{p}\lesssim r^{-\frac{p-1}{p}d}\). In order to control the terms in line (52), we shall establish the following estimates for \(n\in {\mathbb {N}}\)

*n*in (52) converges and gives (37).

*v*of the constant coefficient Eq. (27) is classical: We already argued that \(\Vert \nabla v\Vert _{p}\lesssim r^{-\frac{p-1}{p}d}\); by the estimate on the support of

*g*in (48) we have that

*v*is harmonic in \(\{|x|\ge r\}\) and that it has vanishing flux \(\int _{|x|=r}x\cdot \nabla v=0\). It thus decays as \(|\nabla v(x)|\lesssim |x|^{-d} r^{d-\frac{d}{p}} \Vert \nabla v\Vert _{p}\) for \(|x|\ge 2r\), which in particular yields (53). We now turn to (54) and to this purpose rewrite the Eq. (28) for \({\tilde{v}}_{jk}\) as

*w*the corresponding Lax–Milgram solution of (43). By integration by parts, we deduce from (43) and (57) that \(\int h\cdot \nabla {\tilde{v}}_m dx=\int {\tilde{g}}_m\cdot \nabla w\,dx\). By the support condition on \({\tilde{g}}_m\) this yields

*h*we have that

*w*is

*a*-harmonic in \(\{|x|\le 2^n r\}\). Since \(m<n\), we may use (45) applied to

*w*in form of

*F*is again controlled in the sense of (37). The proof is mostly analogous to the previous one; we again rewrite

*F*as in (49) with some

*g*satisfying (48). Starting from the representation (31), one derives an analogue of estimate (50) reading

### 3.4 Sufficient conditions for the mean value property in terms of linear functionals of the corrector

### Proof of Lemma 4

*h*by extension à la Hahn-Banach. \(\square \)

## 4 Proof of main result

### Proof of Theorem 1

**Proof of Assertion (i)**.

Consider the functionals \(\{F_{n}\}_{n=1,\ldots ,N}\) and their rescalings \(\{F_{n,R}\}_{n=1,\ldots ,N;R\;\text {dyadic}}\) constructed in Lemma 4. Let us abbreviate \(F_{n,R}\nabla (\phi ,\sigma )\) as \(F_{n,R}\). We would like to apply concentration of measure to these functionals.

*a*-harmonic functions down to scale

*r*. By Lemma 4 this assumption may be reduced to the smallness assumption (39) for our functionals \(F_{m,R}\) on scales \(R\ge r\), so that Lemma 3 becomes applicable under the assumption (39): Let

*q*be related to \(\beta \) through (24) and let

*p*be related to

*q*through (34). By the smallness assumption on \(\beta \) in our theorem (cf. (13)), we deduce that (33) holds. By scaling, our functionals \(F_{n,r}\) satisfy the estimate (32) up to a universal constant factor. Furthermore, by ergodicity the property (36) holds for \(\langle \cdot \rangle \)-almost every coefficient field

*a*(regarding \(\sigma \), this result has been shown in [11, Lemma 1]; for \(\phi \), it is classical but may also be found in [11]). Thus, the estimate (37) holds for \(F_{n,r}\) under the assumption (39), i.e. there exists a constant \(C_0\) only depending on

*d*, \(\lambda \), and \(\beta \), such that for any \(n=1,\ldots ,N\) and any radius

*r*the implication

*a*.

To apply concentration of measure in the form of Proposition 1 to some functional *F*, we however need an unconditional bound on the Malliavin derivative (cf. (21) respectively (25)).

*c*is some small universal constant). This yields

*finite*radius \(r_0\) which is minimal with the property

*quantitative*estimate on \(r_0\). To this purpose we now consider the auxiliary variable

*c*being a small universal constant)

*r*, the above holds without the lower restriction on

*M*:

*r*replaced by

*R*and summing over the finite index set \(n=1,\ldots ,N\) and all dyadic \(R\ge r\) we obtain

*M*is immaterial since \(\bar{F}_r\le \frac{1}{C_0}\le 1\). Using \(\langle \bar{F}_r\rangle =\int _0^\infty \langle I({\bar{F}}_r\ge M)\rangle dM\), this yields the following quantification of \(\lim _{r\uparrow \infty }\langle {\bar{F}}_r\rangle =0\):

*r*, this yields the desired

*F*from (14) into the list of finitely many functionals \(F_1,\ldots ,F_N\), say, as the last functional \(F_N=F\), and then to specify the above to \(n=N\). We note that for

*q*related to \(\beta \) through (24) and

*p*related to

*q*through (34) one has \(\frac{p}{p-1}=\frac{2d}{d+\beta }\), i.e. (14) entails (32). Note that by adjusting the constants, (15) is trivial for \(r\lesssim 1\), so that we obtain (15) over the whole range \(r\ge 0\).

**Proof of Assertion (ii)**.

*not*assume \(r_*<\infty \). In order to establish (17), it is enough to show for a given dyadic \(r_0\ge 1\) that

*f*(

*z*) grows sub-algebraically. For the l. h. s. of (78) we note

*Q*of “level

*r*” (that is, of side length 2

*r*) of the cube \((-R,R)^d\). In other words, on such a sub-cube

*Q*, \((\phi ,\sigma )_r=-\int _{Q}(\phi ,\sigma )dx\). With this language, we may rewrite the first r. h. s. term of (80) as

*Q*we obtain

*Q*of \((-R,R)^d\) of level

*r*we introduce the \(N=2^d\) linear functionals \(F_{Q,n}\) as an extension of

*Q*, and which satisfy the desired boundedness property (14) restricted to gradient fields (which is no issue because of Hahn–Banach extension) and translated (which will be no issue because of stationarity), that is,

- a (dyadic) \(r\in [2r_1,R]\), a sub-cube
*Q*of \((-R,R)^d\) of level*r*, and an index \(n=1,\ldots ,2^d\) such that \((F_{Q,n}\nabla (\phi ,\sigma ))^2\gtrsim (\frac{R}{r})^2g(\frac{R}{r})(\frac{r_0}{R})^\beta f(\frac{R}{r_0})\). In view of the boundedness condition (82) and stationarity, we may apply (15) with*F*replaced by \(F_{Q,n}\) and \(M^2\) replaced by \((\frac{R}{r})^2g(\frac{R}{r})(\frac{r_0}{R})^\beta f(\frac{R}{r_0})\). This*M*is admissible in the sense of \(M\lesssim 1\) because by (79) we have \((\frac{R}{r})^2g(\frac{R}{r})(\frac{r_0}{R})^\beta f(\frac{R}{r_0}) \le (\frac{R}{r_1})^2(\frac{r_0}{R})^\beta f(\frac{R}{r_0})\sim 1\). Hence the probability of each single of this events is estimated as followsSince$$\begin{aligned}&\left\langle I\left( (F_{Q,n}\nabla (\phi ,\sigma ))^2 \ge \frac{1}{C}\left( \frac{R}{r}\right) ^2g\left( \frac{R}{r}\right) \left( \frac{r_0}{R}\right) ^\beta f\left( \frac{R}{r_0}\right) \right) \right\rangle \\&\quad \lesssim \exp \left( -\frac{1}{C}\left( \frac{R}{r}\right) ^{2-\beta }g\left( \frac{R}{r}\right) f\left( \frac{R}{r_0}\right) r_0^\beta \right) . \end{aligned}$$*g*(*z*) decays sub-algebraically in*z*and since \(\beta <2\), this yields the simpler form$$\begin{aligned}&\left\langle I\left( (F_{Q,n}\nabla (\phi ,\sigma ))^2 \ge \frac{1}{C}\left( \frac{R}{r}\right) ^2g\left( \frac{R}{r}\right) \left( \frac{r_0}{R}\right) ^\beta f\left( \frac{R}{r_0}\right) \right) \right\rangle \\&\quad \lesssim \exp \left( -\frac{1}{C}\left( \frac{R}{r}\right) ^{1-\frac{\beta }{2}} f\left( \frac{R}{r_0}\right) r_0^\beta \right) . \end{aligned}$$ -
*or*a (dyadic) \(r\ge 2R\) and an index \(n=1,\ldots ,N\) for which the estimate \((F_{n,r}\nabla (\phi ,\sigma ))^2\gtrsim 1\) holds. By the boundedness property of \(F_{n,r}\), each single of these events is estimated as$$\begin{aligned} \left\langle I\left( (F_{n,r}\nabla (\phi ,\sigma ))^2 \ge \frac{1}{C}\right) \right\rangle \lesssim \exp \left( -\frac{1}{C}r^\beta \right) . \end{aligned}$$

*Q*into account and recalling \(N\lesssim 1\), this implies

*r*in (84) and \(A=\frac{1}{C}f(\frac{R}{r_0})r_0^\beta \), which satisfies \(A\gg 1\) for \(r_0\gg 1\), and using the estimate \(\sum _{r\ge 2R; r\text { dyadic}} \exp (-\frac{1}{C}r^\beta ) \lesssim \exp (-\frac{1}{C} R^\beta )\) (which holds provided that \(R\ge r_0\ge 1\)) for the second sum over

*r*, we obtain

*A*this yields (77). Note that the condition \(r_0\gg 1\) is immaterial after adjusting the constants, as the l. h. s. of (77) is bounded by 1. \(\square \)

## Notes

### Acknowledgements

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