Kantorovich distance in the martingale CLT and quantitative homogenization of parabolic equations with random coefficients

TL;DR

This paper develops a quantitative martingale CLT using Kantorovich distance and applies it to prove polynomial convergence rates in the homogenization of discrete parabolic and elliptic equations with random coefficients.

Contribution

It introduces a new quantitative CLT in terms of Kantorovich distance and applies it to establish explicit polynomial rates of homogenization for random parabolic and elliptic equations.

Findings

Polynomial convergence rate for the averaged heat kernel.

Explicit exponent depending on the dimension.

Homogenization results for both parabolic and elliptic equations.

Abstract

The article begins with a quantitative version of the martingale central limit theorem, in terms of the Kantorovich distance. This result is then used in the study of the homogenization of discrete parabolic equations with random i.i.d. coefficients. For smooth initial condition, the rescaled solution of such an equation, once averaged over the randomness, is shown to converge polynomially fast to the solution of the homogenized equation, with an explicit exponent depending only on the dimension. Polynomial rate of homogenization for the averaged heat kernel, with an explicit exponent, is then derived. Similar results for elliptic equations are also presented.