Quantitative stochastic homogenization: local control of homogenization error through corrector

TL;DR

This paper establishes a deterministic link between the sublinear growth of correctors and the decay rate of homogenization error in linear elliptic equations with random coefficients, enhancing understanding of local error control.

Contribution

It introduces an expanded notion of corrector involving scalar and vector potentials, demonstrating their sublinear growth directly controls the homogenization error decay rate.

Findings

Sublinear growth of correctors implies decay rate of homogenization error.

Deterministic relationship between corrector behavior and Green's function decay.

Enhanced local control of homogenization error through corrector analysis.

Abstract

This note addresses the homogenization error for linear elliptic equations in divergence-form with random stationary coefficients. The homogenization error is measured by comparing the quenched Green's function to the Green's function belonging to the homogenized coefficients, more precisely, by the (relative) spatial decay rate of the difference of their second mixed derivatives. The contribution of this note is purely deterministic: It uses the expanded notion of corrector, namely the couple of scalar and vector potentials $(\phi,\sigma)$, and shows that the rate of sublinear growth of $(\phi,\sigma)$ at the points of interest translates one-to-one into the decay rate.