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

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