A user-friendly condition for exponential ergodicity in randomly switched environments

Abstract

We consider random switching between finitely many vector fields leaving positively invariant a compact set. Recently, Li, Liu and Cui showed in [12] that if one the vector fields has a globally asymptotically stable (G.A.S.) equilibrium from which one can reach a point satisfying a weak Hörmander-bracket condition, then the process converges in total variation to a unique invariant probability measure. In this note, adapting the proof in [12] and using results of [5], the assumption of a G.A.S. equilibrium is weakened to the existence of an accessible point at which a barycentric combination of the vector fields vanishes. Some examples are given which demonstrate the usefulness of this condition.