L 2 hypocoercivity, deviation bounds, hitting times and Lyapunov functions
Annales mathématiques Blaise Pascal, Volume 29 (2022) no. 2, pp. 295-306.

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. A french version is available [14].

On montre que, pour un semi-groupe de Markov, l’hypocoercivité L 2 (c’est-à-dire la contractivité d’une norme L 2 modifiée) implique des inégalités de concentration quantitatives et l’intégrabilité exponentielle des temps d’atteinte des ensembles de mesure positive. D’autre part, pour les diffusions et sous une hypothèse forte d’hypoellipticité, on établit que l’hypocoercivité L 2 implique l’existence d’une fonction de Lyapunov pour le générateur associé. Une version en français est disponible [14].

Published online:
DOI: 10.5802/ambp.414
Classification: 60J25, 35F15, 35H10
Keywords: Hypocoercivité, fonctions de Lyapunov
Mot clés : Hypocoercivity, Lyapunov functions
Pierre Monmarché 1

1 Sorbonne Université Laboratoire Jacques-Louis Lions 4 place Jussieu 75011 Paris, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Pierre Monmarché. $L^2$ hypocoercivity, deviation bounds, hitting times and Lyapunov functions. Annales mathématiques Blaise Pascal, Volume 29 (2022) no. 2, pp. 295-306. doi : 10.5802/ambp.414. https://ambp.centre-mersenne.org/articles/10.5802/ambp.414/

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