Quantitative homogenization in a balanced random environment

TL;DR

This paper establishes quantitative rates for homogenization and the quenched central limit theorem for random walks in balanced random environments, advancing understanding of their ergodic and probabilistic properties.

Contribution

It provides the first quantitative homogenization results and algebraic convergence rates for non-divergence form difference operators in balanced random environments.

Findings

Quantified ergodicity of the environment from the particle's perspective.

Proved algebraic rate of convergence for the quenched CLT.

Established algebraic homogenization rates for elliptic and parabolic difference operators.

Abstract

We consider discrete non-divergence form difference operators in a random environment and the corresponding process--the random walk in a balanced random environment in $\mathbb{Z}^d$ with a finite range of dependence. We first quantify the ergodicity of the environment from the point of view of the particle. As a consequence, we quantify the quenched central limit theorem of the random walk with an algebraic rate. Furthermore, we prove an algebraic rate of convergence for the homogenization of the Dirichlet problems for both elliptic and parabolic non-divergence form difference operators.