International audienceWe consider a discrete elliptic equation with random coefficients $A$, which (to fix ideas) are identically distributed and independent from grid point to grid point $x\in\mathbb{Z}^d$. On scales large w.\ r.\ t.\ the grid size (i.\ e.\ unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. These symmetric ''homogenized'' coefficients $A_{hom}$ are characterized by % $$ \xi\cdot A_{hom}\xi\;=\;\langle\left((\xi+\nabla\phi)\cdot A(\xi+\nabla\phi)\right)(0)\rangle, \quad\xi\in\mathbb{R}^d, $$ % where the random field $\phi$ is the unique stationary solution of the ''corrector problem'' % $$ -\nabla\cdot A(\xi+\nabla\phi)\;=\;0 $$ % and $\langle\cdot\rangle$ denotes the ensemble average. \medskip It is known (''by ergodicity'') that the above ensemble average of the energy density $e=(\xi+\nabla\phi)\cdot A(\xi+\nabla\phi)$, which is a stationary random field, can be recovered by a system average. We quantify this by proving that the variance of a spatial average of $e$ on length scales $L$ is estimated as follows: % $$ {\rm var}\left[\sum_{x\in\mathbb{Z}^d}\eta_L(x)\,e(x)\right] \;\lesssim\;L^{-d}, $$ % where the averaging function (i.\ e.\ $\sum_{x\in\mathbb{Z}^d}\eta_L(x)=1$, ${\rm supp}\eta_L\subset[-L,L]^d$) has to be smooth in the sense that $|\nabla\eta_L|\lesssim L^{-1}$. In two space dimensions (i.\ e.\ $d=2$), there is a logarithmic correction. \medskip In other words, smooth averages of the energy density $e$ behave like as if $e$ would be independent from grid point to grid point (which it is not for $d>1$). This result is of practical significance, since it allows to estimate the error when numerically computing $A_{hom}$

We study the effective large-scale behavior of discrete elliptic equations on the lattice Z d with random coefficients. The theory of stochastic homogenization relates the random but stationary field of coefficients with a deterministic matrix of effective coefficients. This is done via the corrector problem, which can be viewed as a highly degenerate elliptic equation on the infinite-dimensional space of admissible coefficient fields. In this contribution we develop quantitative methods for the corrector problem assuming that the ensemble of coefficient fields satisfies a spectral gap estimate w. r. t. a Glauber dynamics. As a main result we prove an optimal estimate for the decay in time of the parabolic equation associated to the corrector problem (i. e. for the "random environment as seen from a random walker"). As a corollary we obtain existence and moment bounds for stationary correctors (in dimension d > 2) and optimal estimates for regularized versions of the corrector (in dimensions d ≥ 2). We also give a self-contained proof for a new estimate on the gradient of the parabolic, variable-coefficient Green's function, which is a crucial analytic ingredient in our method.As an application, we study the approximation of the homogenized coefficients via a representative volume element. The approximation introduces two types of errors. Based on our quantitative methods, we develop an error analysis that gives optimal bounds in terms of scaling in the size of the representative volume element -even for large ellipticity ratios.

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This paper is the companion article to [Ann. Probab. 39 (2011) 779-856]. We consider a discrete elliptic equation on the d-dimensional lattice Z d with random coefficients A of the simplest type: They are identically distributed and independent from edge to edge. On scales large w.r.t. the lattice spacing (i.e., unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. This symmetric "homogenized" matrix A hom = a hom Id is characterized by ξ · A hom ξ = (ξ + ∇φ) · A(ξ + ∇φ) for any direction ξ ∈ R d , where the random field φ (the "corrector") is the unique solution of −∇ * · A(ξ + ∇φ) = 0 in Z d such that φ(0) = 0, ∇φ is stationary and ∇φ = 0, · denoting the ensemble average (or expectation).In order to approximate the homogenized coefficients A hom , the corrector problem is usually solved in a box Q L = [−L, L) d of size 2L with periodic boundary conditions, and the space averaged energy on Q L defines an approximation A L of A hom . Although the statistics is modified (independence is replaced by periodic correlations) and the ensemble average is replaced by a space average, the approximation A L converges almost surely to A hom as L ↑ ∞. In this paper, we give estimates on both errors. To be more precise, we do not consider periodic boundary conditions on a box of size 2L, but replace the elliptic operator by T −1 − ∇ * · A∇ with (typically) T ∼ L 2 , as standard in the homogenization literature. We then replace the ensemble average by a space average on Q L , and estimate the overall error on the homogenized coefficients in terms of L and T .

The qualitative theory of stochastic homogenization of uniformly elliptic linear (but possibly non-symmetric) systems in divergence form is well-understood. Quantitative results on the speed of convergence and on the error in the representative volume method, like those recently obtained by the authors for scalar equations, require a type of stochastic regularity theory for the corrector (e.g., higher moment bounds). One of the main insights of the very recent work of Armstrong and Smart is that one should separate these error estimates, which require strong mixing conditions in order to yield the best rates possible, from the (large scale) regularity theorỳ a la Avellaneda & Lin for a-harmonic functions that should hold under milder mixing conditions. In this paper, we establish an intrinsinc C 1,1 -version of the improved regularity theory for non-symmetric random systems, that is qualitative in the sense it yields a new Liouville theorem under mere ergodicity, and that is quantifiable in the sense that it holds under a condition that has high stochastic integrabiliity provided the coefficients satisfy quantitative mixing assumptions. We introduce such a class of quantitative mixing condition, that allows for arbitrarily slow-decaying correlations, and under which we derive a new family of optimal estimates in stochastic homogenization.