Electronic Communications in Probability

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

Michel Benaïm, Tobias Hurth, and Edouard Strickler

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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.

Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 44, 12 pp.

Dates
Received: 13 March 2018
Accepted: 28 June 2018
First available in Project Euclid: 25 July 2018

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1532505675

Digital Object Identifier
doi:10.1214/18-ECP148

Mathematical Reviews number (MathSciNet)
MR3841405

Zentralblatt MATH identifier
1397.60106

Subjects
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)

Keywords
piecewise deterministic Markov processes random switching Hörmander-bracket conditions ergodicity stochastic persistence

Rights
Creative Commons Attribution 4.0 International License.

Citation

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


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References

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