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| HAL : hal-00431118, version 2 |
| arXiv : 0911.1973 |
| DOI : 10.1214/10-AAP757 |
| Fiche détaillée |  Récupérer au format |
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| Annals of Applied Probability 21, 6 (2011) 2263-2314 |
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| Versions disponibles : | v1 (10-11-2009) | v2 (02-08-2010) |
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| Limit theorems for Markov processes indexed by continuous time Galton-Watson trees |
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| Vincent Bansaye 1Jean-François Delmas 2 |
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| (12/2011) |
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| We study the evolution of a particle system whose genealogy is given by a supercritical continuous time Galton-Watson tree. The particles move independently according to a Markov process and when a branching event occurs, the offspring locations depend on the position of the mother and the number of offspring. We prove a law of large numbers for the empirical measure of individuals alive at time $t$. This relies on a probabilistic interpretation of its intensity by mean of an auxiliary process. This latter has the same generator as the Markov process along the branches plus additional branching events, associated with jumps of accelerated rate and biased distribution. This comes from the fact that choosing an individual uniformly at time $t$ favors lineages with more branching events and larger offspring number. The central limit theorem is considered on a special case. Several examples are developed, including applications to splitting diffusions, cellular aging, branching Lévy processes and ancestral lineages. |
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| 1 : | Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP) |
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Polytechnique - X – CNRS : UMR7641 |
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| 2 : | Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS) |
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Ecole des Ponts ParisTech |
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| 3 : | Laboratoire Paul Painlevé (LPP) |
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CNRS : UMR8524 – Université Lille 1 - Sciences et Technologies |
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| Domaine | : | Mathématiques/Probabilités |
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| Branching Markov process – Branching diffusion – Limit theorems – Size biased reproduction distribution |
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| Liste des fichiers attachés à ce document : | ||||||||||
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| hal-00431118, version 2 | |
| http://hal.archives-ouvertes.fr/hal-00431118 | |
| oai:hal.archives-ouvertes.fr:hal-00431118 | |
| Contributeur : Viet Chi Tran | |
| Soumis le : Lundi 2 Août 2010, 13:29:38 | |
| Dernière modification le : Vendredi 25 Novembre 2011, 00:38:14 | |
