2016/04/21 by Patrick Cattiaux, Arnaud Guillin, Cattiaux, Patrick +1 · 1 citation
Mathematics · Physics and Astronomy · #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.1604.06336
openalex publication_date 2016/04/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The use of Lyapunov conditions for proving functional inequalities was\ninitiated in [5]. It was shown in [4, 30] that there is an equivalence between\na Poincar 'e inequality, the existence of some Lyapunov function and the\nexponential integrability of hitting times. In the present paper, we close the\nscheme of the interplay between Lyapunov conditions and functional inequalities\nby bullet showing that strong functional inequalities are equivalent to\nLyapunov type conditions; bullet showing that these Lyapunov conditions are\ncharacterized by the finiteness of generalized exponential moments of hitting\ntimes. We also give some complement concerning the link between Lyapunov\nconditions and in-tegrability property of the invariant probability measure and\nas such transportation inequalities , and we show that some "unbounded Lyapunov\nconditions" can lead to uniform ergodicity, and coming down from infinity\nproperty.\n