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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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

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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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)

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

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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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  • [1] Y. Bakhtin and T. Hurth, Invariant densities for dynamical systems with random switching, Nonlinearity 25 (2012), no. 10, 2937–2952.
  • [2] Y. Bakhtin, T. Hurth, S. D. Lawley, and J.C. Mattingly, Smooth invariant densities for random switching on the torus, Nonlinearity 31 (2018), no. 4, 1331.
  • [3] M. Benaïm, Stochastic persistence, (2018), arXiv:1806.08450.
  • [4] M. Benaïm, F. Colonius, and R. Lettau, Supports of invariant measures for piecewise deterministic Markov processes, Nonlinearity 30 (2017), no. 9, 3400–3418.
  • [5] M. Benaïm, S. Le Borgne, F. Malrieu, and P. A. Zitt, Qualitative properties of certain piecewise deterministic Markov processes, Ann. Inst. Henri Poincaré Probab. Stat. 51 (2015), no. 3, 1040–1075.
  • [6] M. Benaïm, S. Le Borgne, F. Malrieu, and P.-A. Zitt, Quantitative ergodicity for some switched dynamical systems, Electron. Commun. Probab. 17 (2012), no. 56, 14.
  • [7] M. Benaïm and C. Lobry, Lotka Volterra in fluctuating environment or “how switching between beneficial environments can make survival harder”, Ann. Appl. Probab. 26 (2016), no. 6, 3754–3785.
  • [8] M. Benaïm and E. Strickler, Random switching between vector fields having a common zero, (2017), arXiv:1702.03089.
  • [9] B. Cloez and M. Hairer, Exponential ergodicity for Markov processes with random switching, Bernoulli 21 (2015), 505–536.
  • [10] M. H. A. Davis, Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models, J. Roy. Statist. Soc. Ser. B 46 (1984), no. 3, 353–388.
  • [11] A. Lajmanovich and J. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci. 28 (1976), no. 3/4, 221–236.
  • [12] D. Li, S. Liu, and J. Cui, Threshold dynamics and ergodicity of an SIRS epidemic model with markovian switching, Journal of Differential Equations 263 (2017), no. 12, 8873–8915.
  • [13] D. Nualart, The Malliavin calculus and related topics, second ed., Probability and its Applications (New York), Springer-Verlag, Berlin, 2006.
  • [14] H. J. Sussmann and V. Jurdjevic, Controllability of nonlinear systems, J. Differential Equations 12 (1972), 95–116.