[Hypocoercivité , inégalité de concentration, temps d’atteinte et fonctions de Lyapunov]
On montre que, pour un semi-groupe de Markov, l’hypocoercivité (c’est-à-dire la contractivité d’une norme 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 implique l’existence d’une fonction de Lyapunov pour le générateur associé. Une version en français est disponible [14].
We establish that, for a Markov semi-group, hypocoercivity, i.e. contractivity for a modified 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 hypocoercivity implies the existence of a Lyapunov function for the generator. A french version is available [14].
Keywords: Hypocoercivité, fonctions de Lyapunov
Mot clés : Hypocoercivity, Lyapunov functions
@article{AMBP_2022__29_2_295_0,
author = {Monmarch\'e, Pierre},
title = {$L^2$ hypocoercivity, deviation bounds, hitting times and {Lyapunov} functions},
journal = {Annales math\'ematiques Blaise Pascal},
pages = {295--306},
publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal},
volume = {29},
number = {2},
year = {2022},
doi = {10.5802/ambp.414},
language = {en},
url = {http://www.numdam.org/articles/10.5802/ambp.414/}
}
TY - JOUR AU - Monmarché, Pierre TI - $L^2$ hypocoercivity, deviation bounds, hitting times and Lyapunov functions JO - Annales mathématiques Blaise Pascal PY - 2022 SP - 295 EP - 306 VL - 29 IS - 2 PB - Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal UR - http://www.numdam.org/articles/10.5802/ambp.414/ DO - 10.5802/ambp.414 LA - en ID - AMBP_2022__29_2_295_0 ER -
%0 Journal Article %A Monmarché, Pierre %T $L^2$ hypocoercivity, deviation bounds, hitting times and Lyapunov functions %J Annales mathématiques Blaise Pascal %D 2022 %P 295-306 %V 29 %N 2 %I Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal %U http://www.numdam.org/articles/10.5802/ambp.414/ %R 10.5802/ambp.414 %G en %F AMBP_2022__29_2_295_0
Monmarché, Pierre. $L^2$ hypocoercivity, deviation bounds, hitting times and Lyapunov functions. Annales mathématiques Blaise Pascal, Tome 29 (2022) no. 2, pp. 295-306. doi : 10.5802/ambp.414. http://www.numdam.org/articles/10.5802/ambp.414/
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