2937 articles  [english version]
 HAL : hal-00379118, version 2
 arXiv : 0904.4175
 DOI : 10.1214/11-AOP644
 Annals of Probability 40, 3 (2012) 1167-1211
 Versions disponibles : v1 (27-04-2009) v2 (12-07-2010) v3 (07-06-2012)
 A continuum-tree-valued Markov process
 (2012)
 We present a construction of a Lévy continuum random tree (CRT) associated with a super-critical continuous state branching process using the so-called exploration process and a Girsanov's theorem. We also extend the pruning procedure to this super-critical case. Let $\psi$ be a critical branching mechanism. We set $\psi_\theta(\cdot)=\psi(\cdot+\theta)-\psi(\theta)$. Let $\Theta=(\theta_\infty,+\infty)$ or $\Theta=[\theta_\infty,+\infty)$ be the set of values of $\theta$ for which $\psi_\theta$ is a branching mechanism. The pruning procedure allows to construct a decreasing Lévy-CRT-valued Markov process $(\ct_\theta,\theta\in\Theta)$, such that $\mathcal{T}_\theta$ has branching mechanism $\psi_\theta$. It is sub-critical if $\theta>0$ and super-critical if $\theta<0$. We then consider the explosion time $A$ of the CRT: the smaller (negative) time $\theta$ for which $\mathcal{T}_\theta$ has finite mass. We describe the law of $A$ as well as the distribution of the CRT just after this explosion time. The CRT just after explosion can be seen as a CRT conditioned not to be extinct which is pruned with an independent intensity related to $A$. We also study the evolution of the CRT-valued process after the explosion time. This extends results from Aldous and Pitman on Galton-Watson trees. For the particular case of the quadratic branching mechanism, we show that after explosion the total mass of the CRT behaves like the inverse of a stable subordinator with index 1/2. This result is related to the size of the tagged fragment for the fragmentation of Aldous' CRT.
 1 : Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO) Université d'Orléans – CNRS : UMR7349 2 : Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique (CERMICS) INRIA – Ecole des Ponts ParisTech
 Domaine : Mathématiques/Probabilités
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 hal-00379118, version 2 http://hal.archives-ouvertes.fr/hal-00379118 oai:hal.archives-ouvertes.fr:hal-00379118 Contributeur : Romain Abraham <> Soumis le : Lundi 12 Juillet 2010, 11:51:13 Dernière modification le : Jeudi 24 Mai 2012, 10:52:22