Phi-entropy inequalities for diffusion semigroups

Details Phi-entropy inequalities for diffusion semigroups Phi-entropy inequalities for diffusion semigroups

2008-12-03

arxiv.org/abs/0812.0800v3

Journal de Math\'ematiques Pures et Appliqu\'es 93, 4 (2010) 449-473

Mathematics

Probability

31 pages

Scientific paper

We obtain and study new $\Phi$-entropy inequalities for diffusion semigroups, with Poincar\'e 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.

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