Piecewise deterministic Markov process — recent results

¹ Inria Bordeaux Sud-Ouest, team CQFD et Université de Bordeaux, IMB, CNRS UMR 5251, 200, Avenue de la Vieille Tour, 33405 Talence cedex, France.
² Laboratoire de Mathématiques Raphaël Salem, Université de Rouen, Avenue de l’Université, BP 12, 76801 Saint-étienne-du-Rouvray, France.
³ Laboratoire de Probabilités et modèles aléatoires, Université Pierre et Marie Curie, 4, place Jussieu, 75005 Paris, France.
⁴ Université de Rennes 1, Institut de Recherche mathématique de Rennes, CNRS-UMR 6625, Campus de Beaulieu. Bâtiment 22, 35042 Rennes Cedex, France.
⁵ Laboratoire d’Analyse et de Mathématiques Appliquées, Université de Marne-la-Vallée, 5, boulevard Descartes, Cité Descartes - Champs-sur-Marne, 77454 Marne-la-Vallée Cedex 2 France>

Abstract

We give a short overview of recent results on a specific class of Markov process: the Piecewise Deterministic Markov Processes (PDMPs). We first recall the definition of these processes and give some general results. On more specific cases such as the TCP model or a model of switched vector fields, better results can be proved, especially as regards long time behaviour. We continue our review with an infinite dimensional example of neuronal activity. From the statistical point of view, these models provide specific challenges: we illustrate this point with the example of the estimation of the distribution of the inter-jumping times. We conclude with a short overview on numerical methods used for simulating PDMPs.