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Elagage d'un arbre de Lévy - Diffusion aléatoire en milieu Lévy

Abstract : Given a general critical or sub-critical branching mechanism, we define a pruning procedure of the associated Lévy continuum random tree. This pruning procedure is defined by adding some marks on the tree, using Lévy snake techniques. We then prove that the resulting sub-tree after pruning is still a Lévy continuum random tree. This last result is proved using the exploration process that codes the CRT, a special Markov property and martingale problems for exploration processes. We then construct, by coupling, an another pruning procedure which define a fragmentation process on the tree. We compute the family of dislocation measures associated with this fragmentation. In a second work, we consider a one-dimensional diffusion in a stable Lévy environment. We show that the normalized local time process refocused at the bottom of the standard valley with height log t converges in law to a functional of two independent Lévy processes conditioned to stay positive. To prove this result, we show that the law of the standard valley is close to a two-sided Lévy process conditioned to stay positive. We also obtain the limit law of the supremum of the normalized local time.
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https://tel.archives-ouvertes.fr/tel-00444554
Contributor : Guillaume Voisin <>
Submitted on : Wednesday, January 6, 2010 - 5:59:53 PM
Last modification on : Thursday, March 5, 2020 - 6:49:17 PM
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Guillaume Voisin. Elagage d'un arbre de Lévy - Diffusion aléatoire en milieu Lévy. Mathématiques [math]. Université d'Orléans, 2009. Français. ⟨tel-00444554⟩

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