Abstract
Explicit rate of convergence in variance (or more general entropies) is obtained for a class of Piecewise Deterministic Markov Processes such as the TCP process, relying on functional inequalities. A method to establish Poincaré (and more generally Beckner) inequalities with respect to a diffusion-type energy for the invariant law of such hybrid processes is developed.
Highlights
This work is devoted to the study of convergence to equilibrium for a class of Piecewise Deterministic Markov Process (PDMP)
Ergodicity and, speed of convergence to the steady state are studied. As far as this last point is concerned, coupling methods have recently proved efficient in order to get explicit rate of convergence in Wasserstein distances for PDMP
For reversible processes (i.e. when L is symmetric in L2(μ)) there is a strong link between, on the one hand, Wasserstein distances and coupling and, on the other ejp.ejpecp.org hand, variance and functional inequalities; PDMP are not reversible
Summary
This work is devoted to the study of convergence to equilibrium for a class of Piecewise Deterministic Markov Process (PDMP). For reversible processes (i.e. when L is symmetric in L2(μ)) there is a strong link between, on the one hand, Wasserstein distances and coupling and, on the other hand, variance (or entropy) and functional inequalities (see [6, 16, 30]); PDMP are not reversible Their invariant measures usually do not satisfy a Poincaré inequality for Γ, which is non-local, not easy to handle, satisfying no chain rule (see [15] for a case in which such an inequality does hold). Our method can be seen as an hypocoercive method of modified Lyapunov functional (see [38, 24, 9], etc.), it is quite simple In these settings, it is usual to assume a Poincaré inequality (1.7) holds. A perturbative results for Poincaré and log-Sobolev inequalities is stated and proved in an Appendix
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
