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

TL;DR

This paper establishes strong point-wise convergence estimates for the Green's function of discrete elliptic equations with strongly mixing random coefficients to the homogenized Green's function, advancing understanding of homogenization in random media.

Contribution

It provides new point-wise estimates on the difference between the averaged and homogenized Green's functions for strongly mixing random environments.

Findings

Bounded the difference between averaged and homogenized Green's functions.

Demonstrated convergence under strongly mixing conditions.

Extended previous results to point-wise estimates in random environments.

Abstract

Consider a discrete uniformly elliptic divergence form equation on the $d$ dimensional lattice $\Z^d$ with random coefficients. It has previously been shown that if the random environment is translational invariant, then the averaged Green's function together with its first and second differences, are bounded by the corresponding quantities for the constant coefficient discrete elliptic equation. It has also been shown that if the random environment is ergodic, then solutions of the random equation converge under diffusive scaling to solutions of a homogenized elliptic PDE on $\R^d$. In this paper point-wise estimates are obtained on the difference between the averaged Green's function and the homogenized Green's function for certain random environments which are strongly mixing.