TRENDS TO EQUILIBRIUM IN TOTAL VARIATION DISTANCE

Patrick Cattiaux (1, 2) , Arnaud Guillin (3)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 LSP - Laboratoire de Statistique et Probabilités
3 LATP - Laboratoire d'Analyse, Topologie, Probabilités

This paper presents different approaches, based on functional inequalities, to study the speed of convergence in total variation distance of ergodic diffusion processes with initial law satisfying a given integrability condition. To this end, we give a general upper bound "à la Pinsker" enabling us to study our problem firstly via usual functional inequalities (Poincaré inequality, weak Poincaré,... ) and truncation procedure, and secondly through the introduction of new functional inequalities $\Ipsi$. These $\Ipsi$-inequalities are characterized through measure-capacity conditions and $F$-Sobolev inequalities. A direct study of the decay of Hellinger distance is also proposed. Finally we show how a dynamic approach based on reversing the role of the semi-group and the invariant measure can lead to interesting bounds.

Mots clés: total variation; diffusion processes; speed of convergence; Poincaré inequality; logarithmic Sobolev inequality; $F$-Sobolev inequality

Domaines: Probabilités [math.PR]

Patrick Cattiaux, Arnaud Guillin. TRENDS TO EQUILIBRIUM IN TOTAL VARIATION DISTANCE. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2009, 45 (1), pp.117-145. ⟨10.1214/07-AIHP152⟩. ⟨hal-00136779⟩