We consider the long-time behavior of a passive scalar advected by an incompressible velocity field. When the flow is uniformly hyperbolic, it is well known that it is possible to construct special anisotropic Sobolev spaces where the solution 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 setting of stochastic velocity fields one can recover certain averaged decay estimates 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 (under certain conditions on the velocity) the Markov process obtained by the advection equation with a random source term has a unique stationary measure describing the statistics of "ideal" scalar turbulence.