Moment bounds on the corrector of stochastic homogenization of non-symmetric elliptic finite difference equations

TL;DR

This paper establishes optimal bounds on the corrector and its gradient in stochastic homogenization of non-symmetric elliptic finite-difference equations, using Logarithmic Sobolev inequalities without maximum principle techniques.

Contribution

It introduces a new method for bounding the corrector that applies to non-symmetric coefficients and elliptic systems, avoiding the maximum principle.

Findings

Optimal bounds on the corrector and its gradient in dimensions d ≥ 2

Method applicable to non-symmetric coefficients and elliptic systems

Utilizes Logarithmic Sobolev inequalities for ergodicity quantification

Abstract

We consider the corrector equation from the stochastic homogenization of uniformly elliptic finite-difference equations with random, possibly non-symmetric coefficients. Under the assumption that the coefficients are stationary and ergodic in the quantitative form of a Logarithmic Sobolev inequality (LSI), we obtain optimal bounds on the corrector and its gradient in dimensions $d \geq 2$. Similar estimates have recently been obtained in the special case of diagonal coefficients making extensive use of the maximum principle and scalar techniques. Our new method only invokes arguments that are also available for elliptic systems and does not use the maximum principle. In particular, our proof relies on the LSI to quantify ergodicity and on regularity estimates on the derivative of the discrete Green's function in weighted spaces.