Qualitative properties of certain piecewise deterministic Markov processes

Abstract

We study a class of Piecewise Deterministic Markov Processes with state space R d × 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.

Sensitivity to switching rates in stochastically switched odes

“…[7], [5], [4], and [1] all study invariant measures for such processes. Our work shows that the existence of…”

“…[7], [5], [4], and [1] all study invariant measures for such processes. Our work shows that the existence of such invariant measures may depend in a complicated way on the switching rates.…”

The Central Limit Theorem for Random Dynamical Systems

“…The next theorem is partially inspired by the reasoning which can be found in Lemma 2.5 [2]. Since the Markov process {(X (t), ξ(t))} t≥0 is defined with he help of the Markov chain {(x n , ξ n )} n∈N given by (3.1), (3.2) we use Theorem 1 and Theorem 2 in the proof of following theorem.…”

“…There is a substantial literature devoted to the problem the existence of an exponentially attractive invariant measure for piecewisedeterministic Markov processes. In [2] the authors considers the particular situation for random dynamical systems without jumps, ( i.e q θ (x) = x), when Y = R d . Under Hormander type bracket conditions, the authors proves that there exists a unique invariant measure and that the processes converges to equilibrium in the total variation norm.…”

Multiscale Piecewise Deterministic Markov Process in infinite dimension: central limit theorem and Langevin approximation

“…Recently, the asymptotic behavior of PDMPs have been investigated in [4,5,12,37], limit theorems for infinite dimensional PDMPs in [35], control problems in [10,11,22], numerical methods in [33,6], time reversal in [27] and to end up this list with no claim of completeness, estimation of the jump rates for PDMPs in [2]. Hybrid systems are the object of great attention because they offer an accurate description of a large class of phenomena arising in various domains such as physics or biology.…”