Quantitative stochastic homogenization of elliptic equations in nondivergence form

TL;DR

This paper presents a new method for stochastic homogenization of elliptic equations in nondivergence form, providing algebraic error estimates and advancing understanding even for linear equations.

Contribution

It introduces a novel approach and geometric quantity for analyzing stochastic homogenization, with new algebraic error bounds under finite dependence assumptions.

Findings

Deviations from the homogenized limit are proportional to a power of the microscopic scale.

The method applies to both nonlinear and linear elliptic equations.

The approach leverages regularity theory for the Monge-Ampère equation.

Abstract

We introduce a new method for studying stochastic homogenization of elliptic equations in nondivergence form. The main application is an algebraic error estimate, asserting that deviations from the homogenized limit are at most proportional to a power of the microscopic length scale, assuming a finite range of dependence. The results are new even for linear equations. The arguments rely on a new geometric quantity which is controlled in part by adapting elements of the regularity theory for the Monge-Amp\`ere equation.