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| HAL: hal-00012342, version 1 |
| arXiv: math.PR/0510439 |
| Detailed view |  Export this paper |
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| Journal of Functional Analysis 238, 2 (2006) 649-677 |
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| Estimates for the density of a nonlinear Landau process |
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| Hélène Guérin 1Sylvie Méléard 2 |
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| (2006) |
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| The aim of this paper is to obtain estimates for the density of the law of a specific nonlinear diffusion process at any positive bounded time. This process is issued from kinetic theory and is called Landau process, by analogy with the associated deterministic Fokker-Planck-Landau equation. It is not Markovian, its coefficients are not bounded and the diffusion matrix is degenerate. Nevertheless, the specific form of the diffusion matrix and the nonlinearity imply the non-degeneracy of the Malliavin matrix and then the existence and smoothness of the density. In order to obtain a lower bound for the density, the known results do not apply. However, our approach follows the main idea consisting in discretizing the interval time and developing a recursive method. To this aim, we prove and use refined results on conditional Malliavin calculus. The lower bound implies the positivity of the solution of the Landau equation, and partially answers to an analytical conjecture. We also obtain an upper bound for the density, which again leads to an unusual estimate due to the bad behavior of the coefficients. |
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| 1: | Institut de Recherche Mathématique de Rennes (IRMAR) |
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CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne |
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| 2: | Modélisation aléatoire de Paris X (MODAL'X) |
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Université Paris X - Paris Ouest Nanterre La Défense |
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| 3: | Laboratoire Analyse, Géométrie et Application (LAGA) |
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CNRS : UMR7539 – Université Paris XIII - Paris Nord – Université Paris VIII - Vincennes Saint-Denis |
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| Subject | : | Mathematics/Probability |
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| Conditional Malliavin calculus – density estimates – nonlinear Landau process – unbounded coefficients – Fokker-Planck-Landau equation |
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| hal-00012342, version 1 | |
| http://hal.archives-ouvertes.fr/hal-00012342 | |
| oai:hal.archives-ouvertes.fr:hal-00012342 | |
| From: Sylvie Méléard | |
| Submitted on: Thursday, 20 October 2005 15:58:17 | |
| Updated on: Tuesday, 16 February 2010 11:10:50 | |
