Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré

We study the relationship between two classical approaches for quantitative ergodic properties: the first one based on Lyapunov type controls and popularized by Meyn and Tweedie, the second one based on functional inequalities (of Poincaré type). We show that they can be linked through new inequalit...

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Bibliographic Details
Published inJournal of functional analysis Vol. 254; no. 3; pp. 727 - 759
Main Authors Bakry, Dominique, Cattiaux, Patrick, Guillin, Arnaud
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.02.2008
Elsevier
Subjects
Ergodic processes
Hypocoercivity
Lyapunov functions
Poincaré inequalities
Hypocoercivity
Lyapunov functions
Poincaré inequalities
Ergodic processes
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Summary:We study the relationship between two classical approaches for quantitative ergodic properties: the first one based on Lyapunov type controls and popularized by Meyn and Tweedie, the second one based on functional inequalities (of Poincaré type). We show that they can be linked through new inequalities (Lyapunov–Poincaré inequalities). Explicit examples for diffusion processes are studied, improving some results in the literature. The example of the kinetic Fokker–Planck equation recently studied by Hérau and Nier, Helffer and Nier, and Villani is in particular discussed in the final section.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2007.11.002