The genealogy of continuous-state branching processes with immigration |
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Authors: | Amaury Lambert |
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Institution: | (1) Laboratoire de Probabilités, CNRS UMR 7599, Université Pierre et Marie Curie, 175, rue du Chevaleret, F-75013 Paris, France. e-mail: lambert@proba.jussieu.fr, FR |
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Abstract: | Recent works by J.F. Le Gall and Y. Le Jan 15] 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
*, called the genealogy-coding process (GCP). We first show its existence using It?’s synthesis theorem. We then give a pathwise
construction of X
* based on a Lévy process X with no negative jumps that does not drift to +∞ and whose Laplace exponent coincides with the branching mechanism, and an
independent subordinator Y whose Laplace exponent coincides with the 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.
Received: 28 January 2000 / Revised version: 5 February 2001 / Published online: 11 December 2001 |
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