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

Abstract

We provide a proof of stochastic homogenization for random environments with a mean zero, divergence-free drift. We prove that the environment homogenizes weakly in ${\mathit{H}}^1$ if the drift admits a stationary ${\mathit{L}}^2$-integrable stream matrix, and we prove that the two-scale expansion converges strongly in ${\mathit{H}}^1$ if the drift admits a stationary ${\mathit{L}}^{\mathit{d}\vee (2+\mathit{\delta })}$-integrable stream matrix. Additionally, under this stronger integrability assumption, we show that the environment almost surely satisfies a large-scale Hölder regularity estimate and first-order Liouville principle.