Qualitative properties of certain piecewise deterministic Markov processes
Michel Bena\"im (UNINE), St\'ephane Le Borgne (IRMAR), Florent Malrieu, (IRMAR, LMPT), Pierre-Andr\'e Zitt (IMB, LAMA)

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…
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Taxonomy
TopicsMarkov Chains and Monte Carlo Methods · Stochastic processes and financial applications · Mathematical Dynamics and Fractals
