Hypocoercivit\'e $L^2$, in\'egalit\'e de concentration, temps d'atteinte et fonctions de Lyapunov

TL;DR

This paper demonstrates that $L^2$ hypocoercivity for Markov semi-groups leads to quantitative concentration inequalities, exponential hitting time integrability, and the existence of Lyapunov functions under hypoellipticity assumptions.

Contribution

It establishes new links between $L^2$ hypocoercivity, concentration inequalities, hitting times, and Lyapunov functions for Markov processes.

Findings

$L^2$ hypocoercivity implies quantitative deviation bounds.

$L^2$ hypocoercivity ensures exponential integrability of hitting times.

For diffusions, hypocoercivity implies the existence of Lyapunov functions.

Abstract

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. An english translation of the original article in french is provided. ----- On montre que, pour un semi-groupe de Markov, l'hypocoercivit\'e $L^2$ -- c'est-\`a-dire la contractivit\'e d'une norme $L^2$ modifi\'ee -- implique des in\'egalit\'es de concentration quantitatives et l'int\'egrabilit\'e exponentielle des temps d'atteinte des ensembles de mesure positive. D'autre part, pour les diffusions et sous une hypoth\`ese forte d'hypoellipticit\'e, on \'etablit que l'hypocoercivit\'e $L^2$ implique l'existence d'une fonction de Lyapunov pour le g\'en\'erateur associ\'e. Une traduction en anglais est disponible.