Optimal convergence rates in stochastic homogenization in a balanced random environment

TL;DR

This paper establishes optimal convergence rates for stochastic homogenization of non-divergence form operators in balanced i.i.d. environments, providing quantitative results on invariant measures, correctors, and the quenched CLT.

Contribution

It introduces nearly optimal convergence rates and detailed quantitative analysis for stochastic homogenization in balanced random environments across all dimensions.

Findings

Quantitative law of large numbers for the invariant measure

Mixing property of the invariant measure field

Explicit convergence rates for the quenched CLT

Abstract

We consider random walks in a uniformly elliptic, balanced, i.i.d. random environment in the integer lattice $Z^d$ for $d\geq 2$ and the corresponding problem of stochastic homogenization of non-divergence form difference operators. We first derive a quantitative law of large numbers for the invariant measure, which is nearly optimal. A mixing property of the field of the invariant measure is then achieved. We next obtain rates of convergence for the homogenization of the Dirichlet problem for non-divergence form operators, which are generically optimal for $d\geq 3$ and nearly optimal when $d=2$. Furthermore, we establish the existence, stationarity and uniqueness properties of the corrector problem for all dimensions $d\ge 2$. Afterwards, we quantify the ergodicity of the environmental process for both the continuous-time and discrete-time random walks, and as a consequence, we get explicit convergence rates for the quenched central limit theorem of the balanced random walk.