Université Paris 6
Pierre et Marie Curie
Université Paris 7
Denis Diderot

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

The genealogy of continuous-state branching processes with immigration


Code(s) de Classification MSC:

RÚsumÚ: 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 It˘'s synthesis theorem. We then give a pathwise construction of $X^\star$\ based on a LÚvy 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 LÚvy process with no negative jumps.

Mots ClÚs: LÚvy process ; continuous-state branching process with immigration ; genealogy ; height process ; excursion measure

Date: 1999-12-17

Prépublication numéro: PMA-555