Qualitative properties of certain piecewise deterministic Markov processes

TL;DR

This paper investigates the qualitative properties of a class of piecewise deterministic Markov processes, focusing on their construction, support, long-term behavior, and conditions for unique invariant measures.

Contribution

It provides a detailed construction, support characterization, and analyzes long-term behavior, including conditions for uniqueness of invariant measures under bracket conditions.

Findings

Support characterized via differential inclusion solutions

Unique invariant measure under Hormander-type conditions

Multiple invariant measures possible without bracket conditions

Abstract

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 H\"ormander-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.