Quantitative homogenization of degenerate random environments

TL;DR

This paper investigates the conditions under which the environment in a random conductance model becomes diffusive, providing criteria for polynomial moment estimates on the corrector in degenerate random environments.

Contribution

It establishes a simple necessary and sufficient condition for diffusive relaxation in the environment seen by the particle in the model.

Findings

Identifies a key condition for diffusive behavior in the environment

Provides polynomial moment estimates for the corrector

Clarifies the role of environment degeneracy in homogenization

Abstract

We study discrete linear divergence-form operators with random coefficients, also known as the random conductance model. We assume that the conductances are bounded, independent and stationary; the law of a conductance may depend on the orientation of the associated edge. We give a simple necessary and sufficient condition for the relaxation of the environment seen by the particle to be diffusive, in the sense of every polynomial moment. As a consequence, we derive polynomial moment estimates on the corrector.