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| HAL : hal-00021591, version 1 |
| arXiv : math.PR/0603541 |
| Fiche détaillée |  Récupérer au format |
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| Probability Theory and Related Fields 140, 1-2 (2008) 19-40 |
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| Probabilistic approach for granular media equations in the non uniformly convex case. |
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| Patrick Cattiaux 1, 2Arnaud Guillin 3 |
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| (2008) |
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| We use here a particle system to prove a convergence result as well as a deviation inequality for solutions of granular media equation when the confinement potential and the interaction potential are no more uniformly convex. Proof is straightforward, simplifying deeply proofs of Carrillo-McCann-Villani \cite{CMV,CMV2} and completing results of Malrieu \cite{malrieu03} in the uniformly convex case. It relies on an uniform propagation of chaos property and a direct control in Wasserstein distance of solutions starting with different initial measures. The deviation inequality is obtained via a $T_1$ transportation cost inequality replacing the logarithmic Sobolev inequality which is no more clearly dimension free. |
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| 1 : | Modélisation aléatoire de Paris X (MODAL'X) |
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Université Paris X - Paris Ouest Nanterre La Défense |
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| 2 : | Centre de Mathématiques Appliquées (CMAP) |
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CNRS : UMR7641 – Université de Versailles Saint-Quentin-en-Yvelines – Polytechnique - X |
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| 3 : | CEntre de REcherches en MAthématiques de la DEcision (CEREMADE) |
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CNRS : UMR7534 – Université Paris IX - Paris Dauphine |
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| 4 : | 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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| 5 : | Laboratoire de statistique et probabilités |
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Aucune |
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| Processus stochastiques |
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| Domaine | : | Mathématiques/Probabilités |
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| Granular media equation – transportation cost inequality – Logarithmic Sobolev Inequalities – Concentration inequalities. |
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| Liste des fichiers attachés à ce document : | ||||||||||
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| hal-00021591, version 1 | |
| http://hal.archives-ouvertes.fr/hal-00021591 | |
| oai:hal.archives-ouvertes.fr:hal-00021591 | |
| Contributeur : Patrick Cattiaux | |
| Soumis le : Mercredi 22 Mars 2006, 18:47:57 | |
| Dernière modification le : Jeudi 18 Mars 2010, 15:20:51 | |
