Stochastic homogenization with space-time ergodic divergence-free drift

We will discuss the large-scale behavior of diffusions with space-time stationary and ergodic, divergence-free drift. Such processes provide an approximation of transport in rough, incompressible flows and have been used to describe passive advected quantities such as temperature in several contexts. The primary purpose of this talk will be to show that, unlike in the time-independent or time-dependent periodic settings, the time-dependent stochastic case exhibits a fundamentally different behavior. On large-scales, the evolution is characterized by the solution to a stochastic partial differential equation with Stratonovich transport noise. In the absence of spatial ergodicity, the drift is only partially absorbed into the skew-symmetric part of the flux through the use of an appropriately defined stream matrix. This leaves a time-dependent, spatially-homogeneous transport which, for mildly decorrelating fields, converges to a Brownian noise with deterministic covariance in the limit. The results apply to uniformly elliptic, stationary and ergodic environments in which the drift admits a suitably defined stationary and square-integrable stream matrix.