Phi-entropy inequalities for diffusion semigroups

Bolley, François; Gentil, Ivan

Bolley, François; Gentil, Ivan (2010), Phi-entropy inequalities for diffusion semigroups, Journal de Mathématiques Pures et Appliquées, 93, 5, p. 449-473. http://dx.doi.org/10.1016/j.matpur.2010.02.004

Type

Article accepté pour publication ou publié

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http://hal.archives-ouvertes.fr/hal-00344163/en/

Date

2010

Nom de la revue

Journal de Mathématiques Pures et Appliquées

Volume

Numéro

Éditeur

Elsevier

Pages

449-473

Résumé (EN)

We obtain and study new $\Phi$-entropy inequalities for diffusion semigroups, with Poincaré or logarithmic Sobolev inequalities as particular cases. From this study we derive the asymptotic behaviour of a large class of linear Fokker-Plank type equations under simple conditions, widely extending previous results. Nonlinear diffusion equations are also studied by means of these inequalities. The $\Gamma_2$ criterion of D.~Bakry and M. Emery appears as a main tool in the analysis, in local or integral forms.

Mots-clés

Fokker-Planck equation; Diffusion semigroups; Bakry-Emery criterion; Phi-entropies; Poincaré inequality; Functional inequalities; logarithmic Sobolev inequality