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

TL;DR

This paper introduces a simplified condition for exponential ergodicity in randomly switched systems, extending previous results by weakening the assumptions needed for convergence to a unique invariant measure.

Contribution

It generalizes existing ergodicity conditions by replacing the requirement of a globally asymptotically stable equilibrium with the existence of an accessible point where a barycentric combination of vector fields vanishes.

Findings

The new condition ensures exponential ergodicity under broader circumstances.

Examples demonstrate the practical applicability of the weakened condition.

The approach adapts existing proofs and leverages recent theoretical results.

Abstract

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.