Large-scale regularity in stochastic homogenization with divergence-free drift

TL;DR

This paper proves stochastic homogenization and large-scale regularity for environments with divergence-free drift, under integrability conditions, establishing foundational properties like H"older regularity and Liouville principles.

Contribution

It provides a simplified proof of quenched stochastic homogenization for divergence-free drifts with specific integrability conditions, and demonstrates large-scale regularity and Liouville principles.

Findings

Almost sure large-scale H"older regularity

First-order Liouville principle

Homogenization in divergence-free environments

Abstract

We provide a simple proof of quenched stochastic homogenization for random environments with a mean zero, divergence-free drift under the assumption that the drift admits a stationary $L^d$-integrable stream matrix in $d\geq 3$ or an $L^{2+\delta}$-integrable stream matrix in $d=2$. In addition, we prove that the environment almost surely satisfies a large-scale H\"older regularity estimate and first-order Liouville principle.