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| HAL : hal-00019623, version 1 |
| arXiv : math.PR/0602548 |
| Fiche détaillée |  Récupérer au format |
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| ESAIM: Probability and Statistics 12 (2008) 492-504 |
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| Logarithmic Sobolev Inequalities for Inhomogeneous Markov Semigroups |
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| Jean-François Collet 1Florent Malrieu 2 |
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| (2008) |
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| We investigate the dissipativity properties of a class of scalar second order parabolic partial differential equations with time-dependent coefficients. We provide explicit condition on the drift term which ensure that the relative entropy of one particular orbit with respect to some other one decreases to zero. The decay rate is obtained explicitly by the use of a Sobolev logarithmic inequality for the associated semigroup, which is derived by an adaptation of Bakry's $\Gamma-$ calculus. As a byproduct, the systematic method for constructing entropies which we propose here also yields the well-known intermediate asymptotics for the heat equation in a very quick way, and without having to rescale the original equation. |
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| 1 : | Laboratoire Jean Alexandre Dieudonné (JAD) |
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CNRS : UMR6621 – Université de Nice Sophia Antipolis (UNS) |
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| 2 : | Institut de Recherche Mathématique de Rennes (IRMAR) |
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CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – INSA Rennes – Université Rennes II |
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| Processus stochastiques |
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| Domaine | : | Mathématiques/Probabilités |
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| Inhomogeneous Markov process – Logarithmic Sobolev inequality – Relative entropy |
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| Liste des fichiers attachés à ce document : | ||||||||||
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| hal-00019623, version 1 | |
| http://hal.archives-ouvertes.fr/hal-00019623 | |
| oai:hal.archives-ouvertes.fr:hal-00019623 | |
| Contributeur : Florent Malrieu | |
| Soumis le : Vendredi 24 Février 2006, 11:42:53 | |
| Dernière modification le : Lundi 2 Mars 2009, 10:13:11 | |
