Controllability implies mixing. I. Convergence in the total variation metric

This paper is the first part of a project to study the interconnection between the controllability properties of a dynamical system and the large-time asymptotics of trajectories for the associated stochastic system. It is proved that the approximate controllability to a given point and the solid controllability from the same point imply the uniqueness of a stationary measure and exponential mixing in the total variation metric. This result is then applied to random differential equations on a compact Riemannian manifold. In the second part of the project, the solid controllability will be replaced by a stabilisability condition, and it will be proved that this is still sufficient for the uniqueness of a stationary distribution, whereas the convergence to it occurs in the weaker dual-Lipschitz metric.
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