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Qualitative properties of certain piecewise deterministic Markov processes. (English. French summary) Zbl 1325.60123
Summary: 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 solution 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 converge 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.

MSC:
60J25 Continuous-time Markov processes on general state spaces
34A60 Ordinary differential inclusions
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