Fragmentation associated with Levy processes using snake

Romain Abraham (1) , Jean-François Delmas (2)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
2 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique

We consider the height process of a Levy process with no negative jumps, and its associated continuous tree representation. Using Levy snake tools developed by Duquesne and 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 interesting in itself.

Mots clés: Fragmentation; Levy snake; dislocation measure; stable processes; special Markov property

Domaines: Probabilités [math.PR]

Romain Abraham, Jean-François Delmas. Fragmentation associated with Levy processes using snake. Probability Theory and Related Fields, 2008, 141, pp.113-154. ⟨10.1007/s00440-007-0081-2⟩. ⟨hal-00014673v2⟩