Stochastic homogenization of coarse-grained elliptic equations

TL;DR

This paper proves stochastic homogenization for elliptic equations with coefficients that are stationary, ergodic, and satisfy a coarse-grained ellipticity condition, extending understanding of their large-scale behavior.

Contribution

It establishes quenched stochastic homogenization under a novel coarse-grained ellipticity assumption for divergence-form elliptic equations.

Findings

Homogenization holds for coefficients with large-scale boundedness in negative regularity.

A joint integrability condition on symmetric and skew-symmetric parts of coefficients is sufficient.

The results extend previous homogenization theories to broader classes of coefficients.

Abstract

We prove quenched stochastic homogenization for divergence-form elliptic equations, under the assumption that the coefficients are stationary, ergodic, integrable, and satisfy a coarse-grained ellipticity assumption. The ellipticity assumption requires that the coefficients remain bounded in a negative regularity sense on large scales. As a corollary, we recover a sufficient joint integrability condition on the symmetric and skew-symmetric parts of the coefficient field.