L 2 hypocoercivity, deviation bounds, hitting times and Lyapunov functions
[Hypocoercivité L 2 , inégalité de concentration, temps d’atteinte et fonctions de Lyapunov]
Annales mathématiques Blaise Pascal, Tome 29 (2022) no. 2, pp. 295-306.

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].

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].

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

1 Sorbonne Université Laboratoire Jacques-Louis Lions 4 place Jussieu 75011 Paris, France
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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/

[1] Andrieu, Christophe; Durmus, Alain; Nüsken, Nikolas; Roussel, Julien Hypocoercivity of Piecewise Deterministic Markov Process-Monte Carlo (2018) | arXiv

[2] Bakry, Dominique; Cattiaux, Patrick; Guillin, Arnaud 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] Bakry, Dominique; Gentil, Ivan; Ledoux, Michel Analysis and geometry of Markov diffusion operators, Grundlehren der Mathematischen Wissenschaften, 348, Springer, 2014, xx+552 pages | DOI | MR

[4] Benaïm, Michel; Gauthier, Carl-Erik Self-repelling diffusions on a Riemannian manifold, Probab. Theory Relat. Fields, Volume 169 (2017) no. 1-2, pp. 63-104 | DOI | MR

[5] Birrell, Jeremiah; Rey-Bellet, Luc Concentration inequalities and performance guarantees for hypocoercive MCMC samplers (2019) | arXiv

[6] Cattiaux, Patrick 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] Cattiaux, Patrick 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] Cattiaux, Patrick; Guillin, Arnaud Deviation bounds for additive functionals of Markov processes, ESAIM, Probab. Stat., Volume 12 (2008), pp. 12-29 | DOI | MR

[9] Cattiaux, Patrick; Guillin, Arnaud Functional inequalities via Lyapunov conditions, Optimal transportation (London Mathematical Society Lecture Note Series), Volume 413, Cambridge University Press, 2014, pp. 274-287 | MR

[10] Cattiaux, Patrick; Guillin, Arnaud Hitting times, functional inequalities, Lyapunov conditions and uniform ergodicity, J. Funct. Anal., Volume 272 (2017) no. 6, pp. 2361-2391 | DOI | MR

[11] Cattiaux, Patrick; Guillin, Arnaud; Zitt, Pierre-André Poincaré inequalities and hitting times, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 49 (2013) no. 1, pp. 95-118 | DOI | MR

[12] Dolbeault, Jean; Mouhot, Clément; Schmeiser, Christian Hypocoercivity for linear kinetic equations conserving mass, Trans. Am. Math. Soc., Volume 367 (2015) no. 6, pp. 3807-3828 | DOI | MR

[13] Durmus, Alain; Guillin, Arnaud; Monmarché, Pierre Geometric ergodicity of the Bouncy Particle Sampler, Ann. Appl. Probab., Volume 30 (2020) no. 5, pp. 2069-2098 | DOI

[14] Monmarché, Pierre Hypocoercivité L 2 , inégalité de concentration, temps d’atteinte et fonctions de Lyapunov (2019) | arXiv

[15] Talay, Denis 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] Wu, Liming 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] Yosida, Kosaku Functional analysis, Classics in Mathematics, Springer, 1995, xii+501 pages Reprint of the sixth (1980) edition | DOI | MR

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