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Fragmentation associated with Lévy processes using snake
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  • Published: 06 July 2007

Fragmentation associated with Lévy processes using snake

  • Romain Abraham1 &
  • Jean-François Delmas2 

Probability Theory and Related Fields volume 141, pages 113–154 (2008)Cite this article

  • 149 Accesses

  • 19 Citations

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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.

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Author information

Authors and Affiliations

  1. MAPMO, Fédération Denis Poisson, Université d’Orléans, B.P. 6759, 45067, Orléans cedex 2, France

    Romain Abraham

  2. CERMICS, École des Ponts, ParisTech, 6-8 av. Blaise Pascal, Champs-sur-Marne, 77455, Marne La Vallée, France

    Jean-François Delmas

Authors
  1. Romain Abraham
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  2. Jean-François Delmas
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Correspondence to Jean-François Delmas.

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Cite this article

Abraham, R., Delmas, JF. Fragmentation associated with Lévy processes using snake. Probab. Theory Relat. Fields 141, 113–154 (2008). https://doi.org/10.1007/s00440-007-0081-2

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  • Received: 12 December 2005

  • Revised: 09 May 2007

  • Published: 06 July 2007

  • Issue Date: May 2008

  • DOI: https://doi.org/10.1007/s00440-007-0081-2

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Keywords

  • Fragmentation
  • Lévy snake
  • Dislocation measure
  • Stable processes
  • Special Markov property

Mathematics Subject Classification (2000)

  • 60J25
  • 60G57
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