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Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré. (English) Zbl 1146.60058

Authors’ abstract: 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.

MSC:

60J35 Transition functions, generators and resolvents
60J25 Continuous-time Markov processes on general state spaces
60F25 \(L^p\)-limit theorems
31C25 Dirichlet forms
11S99 Algebraic number theory: local fields
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