$L^2$ hypocoercivity, deviation bounds, hitting times and Lyapunov functions

Pierre Monmarché

Published 2019-11-05Version 1

We establish that, for a Markov semi-group, $L^2$ hypocoercivity, i.e. contractivity for a modified $L^2$ norm, implies quantitative deviation bounds for additive functionals of the associated Markov process and exponential integrability of the hitting time of sets with positive measure. Moreover, in the case of diffusion processes and under a strong hypoellipticity assumption, we prove that $L^2$ hypocoercivity implies the existence of a Lyapunov function for the generator.