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Journal Articles Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques Year : 2009

TRENDS TO EQUILIBRIUM IN TOTAL VARIATION DISTANCE

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Abstract

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.
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Dates and versions

hal-00136779 , version 1 (15-03-2007)

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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⟩
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