HITTING TIMES, FUNCTIONAL INEQUALITIES, LYAPUNOV CONDITIONS AND UNIFORM ERGODICITY

Patrick Cattiaux (1) , Arnaud Guillin (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 LMBP - Laboratoire de Mathématiques Blaise Pascal

The use of Lyapunov conditions for proving functional inequalities was initiated in [5]. It was shown in [4, 30] that there is an equivalence between a Poincaré inequality, the existence of some Lyapunov function and the exponential integrability of hitting times. In the present paper, we close the scheme of the interplay between Lyapunov conditions and functional inequalities by • showing that strong functional inequalities are equivalent to Lyapunov type conditions; • showing that these Lyapunov conditions are characterized by the finiteness of generalized exponential moments of hitting times. We also give some complement concerning the link between Lyapunov conditions and in-tegrability property of the invariant probability measure and as such transportation inequalities , and we show that some " unbounded Lyapunov conditions " can lead to uniform ergodicity, and coming down from infinity property.

Mots clés: Lyapunov functions; hitting times; uniform ergodicity; F-Sobolev inequalities; Poincaré inequality; ultracontractivity

Domaines: Probabilités [math.PR]; Analyse fonctionnelle [math.FA]

Patrick Cattiaux, Arnaud Guillin. HITTING TIMES, FUNCTIONAL INEQUALITIES, LYAPUNOV CONDITIONS AND UNIFORM ERGODICITY. Journal of Functional Analysis, 2017, 272 (6), pp.2361-2391. ⟨10.1016/j.jfa.2016.10.003⟩. ⟨hal-01303903⟩