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].
CC-BY 4.0
@article{AMBP_2022__29_2_295_0,
author = {Pierre Monmarch\'e},
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 = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.414/}
}
TY - JOUR AU - Pierre Monmarché 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 - https://ambp.centre-mersenne.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 Pierre Monmarché %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 https://ambp.centre-mersenne.org/articles/10.5802/ambp.414/ %R 10.5802/ambp.414 %G en %F AMBP_2022__29_2_295_0
Pierre Monmarché. $L^2$ 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. https://ambp.centre-mersenne.org/articles/10.5802/ambp.414/
[1] Hypocoercivity of Piecewise Deterministic Markov Process-Monte Carlo (2018) | arXiv
[2] Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré, J. Funct. Anal., Volume 254 (2008) no. 3, pp. 727-759 | DOI | MR
[3] Analysis and geometry of Markov diffusion operators, Grundlehren der Mathematischen Wissenschaften, 348, Springer, 2014, xx+552 pages | DOI | MR
[4] Self-repelling diffusions on a Riemannian manifold, Probab. Theory Relat. Fields, Volume 169 (2017) no. 1-2, pp. 63-104 | DOI | MR
[5] Concentration inequalities and performance guarantees for hypocoercive MCMC samplers (2019) | arXiv
[6] Calcul stochastique et opérateurs dégénérés du second ordre. I. Résolvantes, théorème de Hörmander et applications, Bull. Sci. Math., Volume 114 (1990) no. 4, pp. 421-462 | MR
[7] Calcul stochastique et opérateurs dégénérés du second ordre. II. Problème de Dirichlet, Bull. Sci. Math., Volume 115 (1991) no. 1, pp. 81-122 | MR
[8] Deviation bounds for additive functionals of Markov processes, ESAIM, Probab. Stat., Volume 12 (2008), pp. 12-29 | DOI | MR
[9] Functional inequalities via Lyapunov conditions, Optimal transportation (London Mathematical Society Lecture Note Series), Volume 413, Cambridge University Press, 2014, pp. 274-287 | MR
[10] Hitting times, functional inequalities, Lyapunov conditions and uniform ergodicity, J. Funct. Anal., Volume 272 (2017) no. 6, pp. 2361-2391 | DOI | MR
[11] Poincaré inequalities and hitting times, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 49 (2013) no. 1, pp. 95-118 | DOI | MR
[12] Hypocoercivity for linear kinetic equations conserving mass, Trans. Am. Math. Soc., Volume 367 (2015) no. 6, pp. 3807-3828 | DOI | MR
[13] Geometric ergodicity of the Bouncy Particle Sampler, Ann. Appl. Probab., Volume 30 (2020) no. 5, pp. 2069-2098 | DOI
[14] Hypocoercivité , inégalité de concentration, temps d’atteinte et fonctions de Lyapunov (2019) | arXiv
[15] Stochastic Hamiltonian systems: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme, Markov Process. Relat. Fields, Volume 8 (2002) no. 2, pp. 163-198 | MR
[16] A deviation inequality for non-reversible Markov processes, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 36 (2000) no. 4, pp. 435-445 | MR | Zbl
[17] Functional analysis, Classics in Mathematics, Springer, 1995, xii+501 pages Reprint of the sixth (1980) edition | DOI | MR
Cité par Sources :