Negative regularity mixing for random volume preserving diffeomorphisms

We consider the negative regularity mixing properties of random volume preserving diffeomorphisms on a compact manifold without boundary. We give general criteria so that the associated random transfer operator mixes $H^{-\delta}$ observables exponentially fast in $H^{-\delta}$ (with a deterministic rate), a property that is false in the deterministic setting. The criteria apply to a wide variety of random diffeomorphisms, such as discrete-time iid random diffeomorphisms, the solution maps of suitable classes of stochastic differential equations, and to the case of advection-diffusion by solutions of the stochastic incompressible Navier-Stokes equations on $\mathbb T^2$. In the latter case, we show that the zero diffusivity passive scalar with a stochastic source possesses a unique stationary measure describing "ideal" scalar turbulence. The proof is based on techniques inspired by the use of pseudodifferential operators and anisotropic Sobolev spaces in the deterministic setting.