Strong Convergence to the homogenized limit of elliptic equations with random coefficients II

TL;DR

This paper extends previous results on the convergence of solutions to elliptic equations with random coefficients, now including environments with long-range correlations, by relating them to environments satisfying a Poincaré inequality.

Contribution

It generalizes homogenization convergence results to environments with long-range correlations through convolution techniques.

Findings

Extended convergence rate results to correlated environments.

Established estimates on Green's function differences.

Linked correlated environments to Poincaré inequality satisfying ones.

Abstract

Consider a discrete uniformly elliptic divergence form equation on the $d$ dimensional lattice $\Z^d$ with random coefficients. In [3] rate of convergence results in homogenization and estimates on the difference between the averaged Green's function and the homogenized Green's function for random environments which satisfy a Poincar\'{e} inequality were obtained. Here these results are extended to certain environments with long range correlations. These environments are simply related via a convolution to environments which do satisfy a Poincar\'{e} inequality.