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Probability Theory and Related Fields

, Volume 159, Issue 1–2, pp 357–403 | Cite as

Exit times for an increasing Lévy tree-valued process

  • Romain Abraham
  • Jean-François Delmas
  • Patrick HoscheitEmail author
Article

Abstract

We give an explicit construction of the increasing tree-valued process introduced by Abraham and Delmas using a random point process of trees and a grafting procedure. This random point process will be used in companion papers to study record processes on Lévy trees. We use the Poissonian structure of the jumps of the increasing tree-valued process to describe its behavior at the first time the tree grows higher than a given height, using a spinal decomposition of the tree, similar to the classical Bismut and Williams decompositions. We also give the joint distribution of this exit time and the ascension time which corresponds to the first infinite jump of the tree-valued process.

Keywords

Lévy tree Exit time Tree-valued Markov process  Ascension time Random point measure Spine decomposition 

Mathematics Subject Classification (1991)

60G55 60J25 60J75 60J80 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Romain Abraham
    • 1
  • Jean-François Delmas
    • 2
  • Patrick Hoscheit
    • 2
    Email author
  1. 1.MAPMO, CNRS UMR 7349, Fédération Denis Poisson FR 2964Université d’OrléansOrléans Cedex 2France
  2. 2.Université Paris-Est, CERMICSMarne La ValléeFrance

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