Stochastic homogenization with divergence-free drift

We will first review the role of invariant measures in stochastic homogenization, and explain how the relative success in treating equations without drift, or with a drift that is either the gradient of a stationary field or mean-zero and divergence-free, is due in part to an explicit identification of the invariant measure or to uniform estimates that fail in the general case. We will then restrict our attention to the case of a mean-zero, divergence-free drift. We will prove that such environments homogenize weakly provided the drift admits a stationary, square-integrable stream matrix, thereby providing a simple PDE-based proof of recent optimal results in the discrete setting. Finally, under stronger integrability assumptions on the stream matrix, we will show that the environment satisfies a large-scale Hölder regularity estimate and first-order Liouville principle.