MiS Preprint Repository

We study the effective large-scale behavior of discrete elliptic equations on the lattice ${\mathbb{Z}}^d$ 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 $d\ge 2$). 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.