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

We consider random switching between finitely many vector fields leaving positively invariant a compact set. Recently, Li, Liu and Cui showed 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 H\"ormander-bracket condition, then the process converges in total variation to a unique invariant probability measure. In this note, adapting the proof of Li, Liu and Cui and using results of Bena\"im, Le Borgne, Malrieu and Zitt, 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.
Comment: 14 pages; The article has been accepted for publication in Electronic Communications in Probability