Trends to equilibrium in total variation distance

Cattiaux, Patrick; Guillin, Arnaud

Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 1, pp. 117-145.

Nous étudions ici la vitesse de convergence, pour la distance en variation totale, de diffusions ergodiques dont la loi initiale satisfait une intégrabilité donnée. Nous présentons différentes approches basées sur l’utilisation d’inégalités fonctionnelles. La première étape consiste à donner une borne générale à la Pinsker. Cette borne permet alors d’utiliser, en les combinant à une procedure de troncature, des inégalités usuelles (telles Poincaré ou Poincaré faibles,…). Dans un deuxième temps nous introduisons de nouvelles inégalités appelées ${ℐ}_{\psi }$ que nous caractérisons à l’aide de condition de type capacité-mesure et d’inégalités de type F-Sobolev. Une étude directe de la distance de Hellinger est également proposée. Pour conclure, une approche dynamique basée sur le renversement du rôle du semigroupe de diffusion et de la mesure invariante permet d'obtenir de nouvelles bornes intéressantes.

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 ${ℐ}_{\psi }$. These ${ℐ}_{\psi }$-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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