Green function and invariant measure estimates for nondivergence form elliptic homogenization

TL;DR

This paper provides quantitative estimates for the Green function and invariant measure in stochastic homogenization of nondivergence form elliptic equations, leading to a local CLT and ergodicity results for the associated diffusion.

Contribution

It introduces deterministic estimates valid above a random minimal scale, improving understanding of stochastic homogenization in nondivergence form elliptic equations.

Findings

Quantitative estimates on the Green function and invariant measure

A quenched local CLT for the diffusion process

Quantitative ergodicity estimates for the environmental process

Abstract

We prove quantitative estimates on the the parabolic Green function and the stationary invariant measure in the context of stochasic homogenization of elliptic equations in nondivergence form. We consequently obtain a quenched, local CLT for the corresponding diffusion process and a quantitative ergodicity estimate for the environmental process. Each of these results are characterized by deterministic (in terms of the environment) estimates which are valid above a random, ``minimal'' length scale, the stochastic moments of which we estimate sharply.