Quantitative estimates on the periodic approximation of the corrector in stochastic homogenization

TL;DR

This paper provides quantitative estimates for approximating the corrector in stochastic homogenization of elliptic equations, using a new approach that avoids Green's functions and leverages De Giorgi-Nash-Moser theory.

Contribution

It introduces a novel method for periodic approximation of the corrector in stochastic homogenization, bypassing Green's functions and employing De Giorgi-Nash-Moser theory for $d>2$.

Findings

Quantitative bounds on the periodic approximation of the corrector.

Applicability to diffusion coefficients satisfying a spectral gap estimate.

Extension to dimensions greater than two.

Abstract

In the present contribution we establish quantitative results on the periodic approximation of the corrector equation for the stochastic homogenization of linear elliptic equations in divergence form, when the diffusion coefficients satisfy a spectral gap estimate in probability, and for $d>2$. The main difference with respect to the first part of [Gloria-Otto, arXiv:1409.0801] is that we avoid here the use of Green's functions and more directly rely on the De Giorgi-Nash-Moser theory.