Qualitative properties of certain piecewise deterministic Markov processes

Michel Benaïm (1) , Stéphane Le Borgne (2) , Florent Malrieu (2) , Pierre-André Zitt (3, 4)

1 UNINE - Institut de Mathématiques
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 IMB - Institut de Mathématiques de Bourgogne [Dijon]
4 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

We study a class of Piecewise Deterministic Markov Processes with state space Rd x E where E is a finite set. The continuous component evolves according to a smooth vector field that is switched at the jump times of the discrete coordinate. The jump rates may depend on the whole position of the process. Working under the general assumption that the process stays in a compact set, we detail a possible construction of the process and characterize its support, in terms of the solutions set of a differential inclusion. We establish results on the long time behaviour of the process, in relation to a certain set of accessible points, which is shown to be strongly linked to the support of invariant measures. Under Hörmander-type bracket conditions, we prove that there exists a unique invariant measure and that the processes converges to equilibrium in total variation. Finally we give examples where the bracket condition does not hold, and where there may be one or many invariant measures, depending on the jump rates between the flows.

Mots clés: Piecewise deterministic Markov Process; convergence to equilibrium; differential inclusion; Hörmander bracket condition

Domaines: Probabilités [math.PR]; Systèmes dynamiques [math.DS]

Michel Benaïm, Stéphane Le Borgne, Florent Malrieu, Pierre-André Zitt. Qualitative properties of certain piecewise deterministic Markov processes. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2015, 51 (3), pp.1040-1075. ⟨10.1214/14-AIHP619⟩. ⟨hal-00688920v4⟩