Quantitative ergodicity for some switched dynamical systems

Michel Benaïm (1) , Stéphane Le Borgne (2) , Florent Malrieu (2) , Pierre-André Zitt (3)

1 UNINE - Institut de Mathématiques
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 IMB - Institut de Mathématiques de Bourgogne [Dijon]

We provide quantitative bounds for the long time behavior of a class of Piecewise Deterministic Markov Processes with state space Rd × E where E is a finite set. The continuous component evolves according to a smooth vector field that switches at the jump times of the discrete coordinate. The jump rates may depend on the whole position of the process. Under regularity assumptions on the jump rates and stability conditions for the vector fields we provide explicit exponential upper bounds for the convergence to equilibrium in terms of Wasserstein distances. As an example, we obtain convergence results for a stochastic version of the Morris-Lecar model of neurobiology.

Mots clés: Coupling; Ergodicity; Linear Differential Equations; Piecewise Deterministic Markov Process; Switched dynamical systems; Wasserstein distance

Domaines: Probabilités [math.PR]

Michel Benaïm, Stéphane Le Borgne, Florent Malrieu, Pierre-André Zitt. Quantitative ergodicity for some switched dynamical systems. Electronic Communications in Probability, 2012, 17 (56), pp.1-14. ⟨10.1214/ECP.v17-1932⟩. ⟨hal-00686272v4⟩