Phi-entropy inequalities for diffusion semigroups

We obtain and study new Φ-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–Planck type equations under simple conditions, widely extending previou...

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Bibliographic Details
Published in:Journal de mathématiques pures et appliquées 2010-05, Vol.93 (5), p.449-473
Main Authors: Bolley, François, Gentil, Ivan
Format: Article
Language:English
Subjects:
Diffusion semigroups
Exact sciences and technology
Fokker–Planck equation
General mathematics
General, history and biography
Group theory
Group theory and generalizations
Logarithmic Sobolev inequality
Mathematics
Poincaré inequality
Probability
Sciences and techniques of general use
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Summary:We obtain and study new Φ-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–Planck type equations under simple conditions, widely extending previous results. Nonlinear diffusion equations are also studied by means of these inequalities. The Γ 2 criterion of D. Bakry and M. Emery appears as a main tool in the analysis, in local or integral forms. Nous obtenons et étudions une nouvelle famille d'inégalités Φ-entropiques pour des semigroupes de diffusion, incluant les inégalités de Poincaré et de Sobolev logarithmiques. Nous en déduisons le comportement en temps grand des solutions d'une grande classe d'équations linéaires de type Fokker–Planck, sous de simples conditions. Nous étudions également certaines équations de diffusion nonlinéaires à l'aide de ces inégalités. Cette étude utilise de manière cruciale le critère Γ 2 de D. Bakry et M. Emery, sous des formes locales et intégrales.
ISSN:0021-7824
DOI:10.1016/j.matpur.2010.02.004