Optimal quantitative estimates in stochastic homogenization for elliptic equations in nondivergence form

TL;DR

This paper establishes optimal quantitative estimates for stochastic homogenization of linear elliptic equations in nondivergence form, using concentration inequalities and a novel microscopic regularity theory.

Contribution

It introduces a new $C^{1,1}$ regularity theory at microscopic scales for nondivergence form equations, extending existing divergence form methods.

Findings

Optimal subquadratic growth estimates for correctors

Stretched exponential bounds in probability

Development of a microscopic $C^{1,1}$ regularity theory

Abstract

We prove quantitative estimates for the stochastic homogenization of linear uniformly elliptic equations in nondivergence form. Under strong independence assumptions on the coefficients, we obtain optimal estimates on the subquadratic growth of the correctors with stretched exponential-type bounds in probability. Like the theory of Gloria and Otto \cite{GO1,GO2} for divergence form equations, the arguments rely on nonlinear concentration inequalities combined with certain estimates on the Green's functions and derivative bounds on the correctors. We obtain these analytic estimates by developing a $C^{1,1}$ regularity theory down to microscopic scale, which is of independent interest and is inspired by the $C^{0,1}$ theory introduced in the divergence form case by the first author and Smart \cite{AS2}.