Quantitative ergodicity for some switched dynamical systems
International audience; We provide quantitative bounds for the long time behavior of a class of Piecewise Deterministic Markov Processes with state space Rd × 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. As an example, we obtain convergence results for a stochastic version of the Morris-Lecar model of neurobiology.
Metric properties of the group of area preserving diffeomorphisms
Area preserving cliffeoinorpliisms of the 2-disk which are identity near the boundary form a group D2 wllich can be equipped, usin-g tlhe L2nlorm on its Lie algebra, with a right invariant metric. Witll tllis metric the diameter of D2 is infinite. In this paper we sl-iow that D2 contains quasiisometric embeddings of any finitely generated free group and any finitely generated abelian free group.