Controllability implies mixing I. Convergence in the total variation metric

TL;DR

This paper establishes that certain controllability conditions in dynamical systems guarantee the uniqueness of stationary measures and exponential mixing in total variation, with applications to stochastic differential equations on manifolds.

Contribution

It proves that approximate and solid controllability imply exponential mixing and uniqueness of stationary measures, linking controllability to stochastic system asymptotics.

Findings

Approximate controllability and solid controllability imply exponential mixing.

These controllability conditions ensure the uniqueness of stationary measures.

Application to random differential equations on compact Riemannian manifolds.

Abstract

This paper is the first part of a project devoted to studying the interconnection between 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, we shall replace the solid controllability by a stabilisability condition and prove that it is still sufficient for the uniqueness of a stationary distribution, whereas the convergence to it holds in the weaker dual-Lipschitz metric.