STOCHASTIC HOMOGENIZATION WITH SPACE-TIME ERGODIC DIVERGENCE-FREE DRIFT

We prove that diffusion equations with a space-time stationary and ergodic, divergence -free drift homogenize in law to a deterministic stochastic partial differential equation with Stratonovich transport noise. In the absence of spatial ergodicity, the drift is only partially absorbed into the skewsymmetric part of the flux through the use of an appropriately defined stream matrix. This leaves a time -dependent, spatially -homogenous transport which, for mildly decorrelating fields, converges to a Brownian noise with deterministic covariance in the homogenization limit. The results apply to uniformly elliptic, stationary and ergodic environments in which the drift admits a suitably defined stationary and L2 -integrable stream matrix.