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

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.