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

TL;DR

This paper investigates the convergence of solutions to discrete parabolic equations with random coefficients towards a homogenized limit, extending previous results to environments with slow decay of correlations.

Contribution

It extends existing homogenization convergence results to environments with arbitrarily slow correlation decay, broadening applicability.

Findings

Established convergence rates for equations with slow decay correlations

Extended Green's function estimates to new random environments

Demonstrated robustness of homogenization in broader settings

Abstract

This paper is concerned with the study of solutions to discrete parabolic equations in divergence form with random coefficients, and their convergence to solutions of a homogenized equation. In [11] 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 in which correlations can have arbitrarily small power law decay. Similar results for discrete elliptic equations were obtained in [12].