Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Qualitative properties of certain piecewise deterministic Markov processes

Michel Benaïm, Stéphane Le Borgne, Florent Malrieu, and Pierre-André Zitt

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Abstract

We study a class of piecewise deterministic Markov processes with state space $\mathbb{R}^{d}\times E$ 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.

Résumé

Nous étudions une classe de processus de Markov déterministes par morceaux, sur espace d’états $\mathbb{R}^{d}\times E$ où $E$ est un ensemble fini. La composante continue du processus évolue suivant le flot d’un champ de vecteur, qui change lorsque la composante discrète saute. Les taux de saut peuvent dépendre des deux composantes. Sous l’hypothèse que le processus reste dans un ensemble compact, nous détaillons une construction possible et caractérisons son support en termes de solution d’une inclusion différentielle. Nous étudions ensuite le comportement en temps long, en faisant apparaître un certain ensemble de points accessibles, qui se trouve être fortement lié au support des mesures invariantes. Sous des conditions de type Hörmander sur les crochets de Lie entre les champs de vecteurs, nous montrons qu’il existe une unique mesure invariante vers laquelle le processus converge en variation totale. Nous donnons enfin des exemples où la condition d’unicité n’est pas vérifiée, et où le nombre de mesures invariantes dépend des taux de saut entre les flots.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist. Volume 51, Number 3 (2015), 1040-1075.

Dates
Received: 23 May 2012
Revised: 7 April 2014
Accepted: 12 April 2014
First available in Project Euclid: 1 July 2015

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1435759239

Digital Object Identifier
doi:10.1214/14-AIHP619

Mathematical Reviews number (MathSciNet)
MR3365972

Zentralblatt MATH identifier
1325.60123

Subjects
Primary: 60J99: None of the above, but in this section 34A60: Differential inclusions [See also 49J21, 49K21]

Keywords
Piecewise deterministic Markov process Convergence to equilibrium Differential inclusion Hörmander bracket condition

Citation

Benaïm, Michel; Le Borgne, Stéphane; Malrieu, Florent; Zitt, Pierre-André. Qualitative properties of certain piecewise deterministic Markov processes. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 3, 1040--1075. doi:10.1214/14-AIHP619. https://projecteuclid.org/euclid.aihp/1435759239.


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