Quantitative results on the corrector equation in stochastic homogenization

TL;DR

This paper establishes optimal quantitative estimates for the corrector in stochastic homogenization of linear elliptic equations in divergence form, extending previous discrete results to continuum settings with applications to Poisson random inclusions.

Contribution

It extends prior discrete stochastic homogenization results to continuum equations with spectral gap assumptions, providing optimal estimates and covering Poisson inclusions.

Findings

Optimal estimates for the corrector in continuum stochastic homogenization

Extension of results from discrete to continuum models

Application to Poisson random inclusions

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.