Section: New Results

Long time behavior of piecewise–deterministic Markov processes

Participant : Florent Malrieu.

This is a collaboration with Michel Benaïm (université de Neuchâtel), Stéphane Le Borgne (IRMAR) and Pierre–André Zitt (université de Marne–la–Vallée).

Quantitative ergodicity for some switched dynamical systems

We provide quantitative bounds for the long time behavior of a class of piecewise deterministic Markov processes with state space $R^d\times E$ where $E$ is a finite set. The continuous component evolves according to a smooth vector field that switches at the jump times of the discrete coordinate. The jump rates may depend on the whole position of the process. Under regularity assumptions on the jump rates and stability conditions for the vector fields we provide explicit exponential upper bounds for the convergence to equilibrium in terms of Wasserstein distances [13] . As an example, we obtain convergence results for a stochastic version of the Morris–Lecar model of neurobiology.

On the stability of planar randomly switched systems

Consider the random process $\left(X_t\right)$ solution of $d{X}_t/dt=A\left(I_t\right)X_t$ where $\left(I_t\right)$ is a Markov process on $\{0,1\}$ and $A_0$ and $A_1$ are real Hurwitz matrices on $R^2$. Assuming that there exists $\lambda \in (0,1)$ such that $(1-\lambda )A_0+\lambda A_1$ has a positive eigenvalue, we establish that the norm of $X_t$ may converge to 0 or infinity, depending on the the jump rate of the process $I$. An application to product of random matrices is studied. The paper [29] can be viewed as a probabilistic counterpart of the paper  [36] by Baldé, Boscain and Mason.

Qualitative properties of certain piecewise deterministic Markov processes

We study a class of piecewise deterministic Markov processes with state space $R^m\times E$ where $E$ is a finite set. The continous component evolves according to a smooth vector field that it 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 [30] .