Displaying similar documents to “The geometry of Markov diffusion generators”

Prékopa–Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport

Dario Cordero-Erausquin, Robert J. McCann, Michael Schmuckenschläger (2006)

Annales de la faculté des sciences de Toulouse Mathématiques

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We investigate Prékopa-Leindler type inequalities on a Riemannian manifold M equipped with a measure with density e - V where the potential V and the Ricci curvature satisfy Hess x V + Ric x ≥ λ I for all x ∈ M , with some λ ∈ ℝ . As in our earlier work [], the argument uses optimal mass transport on M , but here, with a special emphasis on its connection with Jacobi fields. A key role will be played by the differential equation satisfied by the determinant of a matrix of Jacobi fields. We also present applications of...

From the Prékopa-Leindler inequality to modified logarithmic Sobolev inequality

Ivan Gentil (2008)

Annales de la faculté des sciences de Toulouse Mathématiques

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We develop in this paper an improvement of the method given by S. Bobkov and M. Ledoux in [BL00]. Using the Prékopa-Leindler inequality, we prove a modified logarithmic Sobolev inequality adapted for all measures on ℝ n , with a strictly convex and super-linear potential. This inequality implies modified logarithmic Sobolev inequality, developed in [GGM05, GGM07], for all uniformly strictly convex potential as well as the Euclidean logarithmic Sobolev inequality.

Trends to equilibrium in total variation distance

Patrick Cattiaux, Arnaud Guillin (2009)

Annales de l'I.H.P. Probabilités et statistiques

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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...