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

### The genealogy of continuous-state branching processes with immigration

Auteur(s):

Code(s) de Classification MSC:

• 60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
• 60J30 Processes with independent increments

Résumé: Recent works by J.F. Le Gall and Y. Le Jan [14] have extended the genealogical structure of Galton-Watson processes to continuous-state branching processes (CB). We are here interested in processes with immigration (CBI). The height process $H$\ which contains all the information about this genealogical structure is defined as a simple local time functional of a strong Markov process $X^\star$, called the genealogy-coding process (GCP). We first show its existence using Itô's synthesis theorem. We then give a pathwise construction of $X^\star$\ based on a Lévy process $X$\ with no negative jumps that does not drift to $+\infty$\ and whose Laplace exponent coincides with the branching mechanism, and an independent subordinator $Y$\ whose Laplace exponent coincides with the immigration mechanism. We conclude the construction with proving that the local time process of $H$\ is a CBI-process. As an application, we derive the analogue of the classical Ray-Knight-Williams theorem for a general Lévy process with no negative jumps.

Mots Clés: Lévy process ; continuous-state branching process with immigration ; genealogy ; height process ; excursion measure

Date: 1999-12-17

Prépublication numéro: PMA-555