Self-similar fragmentations derived from the stable tree II: splitting at nodes

Grégory Miermont (1, 2)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

We study a natural fragmentation process of the so-called stable tree introduced by Duquesne and Le Gall, which consists in removing the nodes of the tree according to a certain procedure that makes the fragmentation self-similar with positive index. Explicit formulas for the semigroup are given, and we provide asymptotic results. We also give an alternative construction of this fragmentation, using paths of Levy processes, hence echoing the two alternative constructions of the standard additive coalescent by fragmenting the Brownian continuum random tree or using Brownian paths, respectively due to Aldous-Pitman and Bertoin.

Mots clés: Self-similar fragmentation; stable tree; stable processes

Domaines: Probabilités [math.PR]

Grégory Miermont. Self-similar fragmentations derived from the stable tree II: splitting at nodes. Probability Theory and Related Fields, 2005, 131, pp.341--375. ⟨10.1007/s00440-004-0373-8⟩. ⟨hal-00001551⟩