Electronic Communications in Probability
- Electron. Commun. Probab.
- Volume 23 (2018), paper no. 44, 12 pp.
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 in  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  and using results of , 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.
Electron. Commun. Probab., Volume 23 (2018), paper no. 44, 12 pp.
Received: 13 March 2018
Accepted: 28 June 2018
First available in Project Euclid: 25 July 2018
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Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60J25: Continuous-time Markov processes on general state spaces 34A37: Differential equations with impulses 37A50: Relations with probability theory and stochastic processes [See also 60Fxx and 60G10] 93E15: Stochastic stability 93C30: Systems governed by functional relations other than differential equations (such as hybrid and switching systems)
Benaïm, Michel; Hurth, Tobias; Strickler, Edouard. A user-friendly condition for exponential ergodicity in randomly switched environments. Electron. Commun. Probab. 23 (2018), paper no. 44, 12 pp. doi:10.1214/18-ECP148. https://projecteuclid.org/euclid.ecp/1532505675