Strong Convergence to the Homogenized Limit of Parabolic Equations with Random Coefficients

TL;DR

This paper establishes strong point-wise convergence estimates for solutions of discrete parabolic equations with strongly mixing random coefficients to their homogenized counterparts, extending previous ergodic results.

Contribution

It provides new point-wise estimates on the difference between random and homogenized solutions for strongly mixing environments, advancing homogenization theory.

Findings

Point-wise convergence estimates are derived.

Results apply to strongly mixing random environments.

Convergence is shown under diffusive scaling.

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. It has previously been shown that if the random environment is translational invariant and ergodic, then solutions of the random equation converge under diffusive scaling to solutions of a homogenized parabolic PDE. In this paper point-wise estimates are obtained on the difference between the averaged solution to the random equation and the solution to the homogenized equation for certain random environments which are strongly mixing.