Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### Two coalescents derived from the ranges of stable subordinators

Auteur(s):

Code(s) de Classification MSC:

• 60J30 Processes with independent increments
• 60G18 Self-similar processes

Résumé: Let $M_\alpha$ be the closure of the range of a stable subordinator of index $\alpha\in ]0,1[$. There are two natural constructions of the $M_{\alpha}$'s simultaneously for all $\alpha\in ]0,1[$, so that $M_{\alpha}\subseteq M_{\beta}$ for $0< \alpha < \beta < 1$: one based on the intersection of independent regenerative sets and one based on Bochner's subordination. We compare the corresponding two coalescent processes defined by the lengths of complementary intervals of $[0,1]\backslash M_{1-\rho}$ for $0 < \rho < 1$. In particular, we identify the coalescent based on the subordination scheme with the coalescent recently introduced by Bolthausen and Sznitman.

Mots Clés: fragmentation ; coalescent ; stable subordinator

Date: 1999-09-28

Prépublication numéro: PMA-530