We consider the long-time behavior of a passive scalar advected by an incompressible velocity field. In the dynamical systems literature, if the velocity field is autonomous or time periodic, long-time behavior follows by studying the spectral properties of the transfer operator associated with the finite time flow map. When the flow is uniformly hyperbolic, it is well known that it is possible to construct certain anisotropic Sobolev spaces where the transfer operator becomes quasi-compact with a spectral gap, yielding exponential decay in these spaces.  In the non-autonomous and non-uniformly hyperbolic case this approach breaks down. In this talk, I will discuss how in the stochastic velocity setting one can recover analogous results under expectation using pseudo differential operators to obtain exponential decay of solutions to the transport equation from H^{-\delta} to H^{-\delta} -- a property we call annealed mixing. As a result, we show that the Markov process obtained by considering the advection diffusion equation with a source term has an H^{-\delta} Wasserstein spectral gap, uniform in diffusivity, and that the stationary measure has a unique limit in the zero diffusivity limit.

This is a joint work with Jacob Bedrossian and Patrick Flynn.