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Lyapunov conditions for Super Poincaré inequalities |
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| Authors: | Patrick Cattiaux Feng-Yu Wang Liming Wu |
| Institution: | a Université Paul Sabatier, Institut de Mathématiques, Laboratoire de Statistique et Probabilités, UMR C 5583, 118 route de Narbonne, F-31062 Toulouse Cedex 09, France b Ecole Centrale Marseille et LATP, Université de Provence, Technopole Château-Gombert, 39, rue F. Joliot Curie, 13453 Marseille Cedex 13, France c Université Blaise Pascal, 33, avenue des landais, 63177 Aubières Cedex, France d School of Mathematical Science, Beijing Normal University, Beijing 100875, China e Department of Mathematics, Swansea University, Singleton Park, SA2 8PP, Swansea, UK f Laboratoire de Mathématiques Appliquées, CNRS-UMR 6620, Université Blaise Pascal, 63177 Aubière, France g Department of Mathematics, Wuhan University, 430072 Hubei, China |
| Abstract: | We show how to use Lyapunov functions to obtain functional inequalities which are stronger than Poincaré inequality (for instance logarithmic Sobolev or F-Sobolev). The case of Poincaré and weak Poincaré inequalities was studied in D. Bakry, P. Cattiaux, A. Guillin, Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré, J. Funct. Anal. 254 (3) (2008) 727-759. Available on Mathematics arXiv:math.PR/0703355, 2007]. This approach allows us to recover and extend in a unified way some known criteria in the euclidean case (Bakry and Emery, Wang, Kusuoka and Stroock, …). |
| Keywords: | Ergodic processes Lyapunov functions Poincaré inequalities Super Poincaré inequalities Logarithmic Sobolev inequalities |
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