Exponential mixing by random velocity fields

We consider the question of exponential mixing for random dynamical systems on arbitrary compact manifolds without boundary. We put forward a robust, dynamics-based framework that allows us to construct space-time smooth, uniformly bounded in time, universal exponential mixers. The framework is then applied to the problem of proving exponential mixing in a classical example proposed by Pierrehumbert in 1994, consisting of alternating periodic shear flows with randomized phases. This settles the open problem of proving the existence of a space-time smooth (universal) exponentially mixing incompressible velocity field on a two-dimensional periodic domain, while also providing a toolbox for constructing such smooth universal mixers in all dimensions. This is based on joint work with Alex Blumenthal (Georgia Tech) and Michele Coti Zelati (ICL).