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Probability Theory and Related Fields

, Volume 141, Issue 1–2, pp 113–154 | Cite as

Fragmentation associated with Lévy processes using snake

  • Romain Abraham
  • Jean-François DelmasEmail author
Article

Abstract

We consider the height process of a Lévy process with no negative jumps, and its associated continuous tree representation. Using Lévy snake tools developed by Le Gall-Le Jan and Duquesne-Le Gall, with an underlying Poisson process, we construct a fragmentation process, which in the stable case corresponds to the self-similar fragmentation described by Miermont. For the general fragmentation process we compute a family of dislocation measures as well as the law of the size of a tagged fragment. We also give a special Markov property for the snake which is of its own interest.

Keywords

Fragmentation Lévy snake Dislocation measure Stable processes Special Markov property 

Mathematics Subject Classification (2000)

60J25 60G57 

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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.MAPMO, Fédération Denis PoissonUniversité d’OrléansOrléans cedex 2France
  2. 2.CERMICS, École des Ponts, ParisTechMarne La ValléeFrance

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