Quantitative results on the corrector equation in stochastic homogenization

Abstract

We derive optimal estimates in stochastic homogenization of linear elliptic equations in divergence form in dimensions $d\ge 2$. In previous works we studied the model problem of a discrete elliptic equation on ${\mathbb{Z}}^d$ . Under the assumption that a spectral gap estimate holds in probability, we proved that there exists a stationary corrector field in dimensions $d>2$ and that the energy density of that corrector behaves as if it had finite range of correlation in terms of the variance of spatial averages – the latter decays at the rate of the central limit theorem. In this article we extend these results, and several other estimates, to the case of a continuum linear elliptic equation whose (not necessarily symmetric) coefficient field satisfies a continuum version of the spectral gap estimate. In particular, our results cover the example of Poisson random inclusions.