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Weak logarithmic Sobolev inequalities and entropic convergence |
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| Authors: | P Cattiaux I Gentil A Guillin |
| Institution: | 1. Ecole Polytechnique, CMAP, 91128, Palaiseau Cedex, France 2. Université Paris X Nanterre, Equipe MODAL’X, UFR SEGMI, 200 avenue de la République, 92001, Nanterre Cedex, France 3. Université Paris-Dauphine, CEREMADE, UMR CNRS 7534, Place du Maréchal De Lattre De Tassigny, 75775, Paris Cedex 16, France 4. Ecole Centrale Marseille et LATP UMR CNRS 6632, Centre de Mathematiques et Informatique Technop?le Chateau-Gombert, 39, rue F. Joliot Curie, 13453, Marseille Cedex 13, France |
| Abstract: | In this paper we introduce and study a weakened form of logarithmic Sobolev inequalities in connection with various others functional inequalities (weak Poincaré inequalities, general Beckner inequalities, etc.). We also discuss the quantitative behaviour of relative entropy along a symmetric diffusion semi-group. In particular, we exhibit an example where Poincaré inequality can not be used for deriving entropic convergence whence weak logarithmic Sobolev inequality ensures the result. |
| Keywords: | Logarithmic Sobolev inequalities Concentration inequalities Entropy |
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