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The genealogy of continuous-state branching processes with immigration
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  • Published: January 2002

The genealogy of continuous-state branching processes with immigration

  • Amaury Lambert1 

Probability Theory and Related Fields volume 122, pages 42–70 (2002)Cite this article

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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 Itô’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.

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Authors and Affiliations

  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

    Amaury Lambert

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  1. Amaury Lambert
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Received: 28 January 2000 / Revised version: 5 February 2001 / Published online: 11 December 2001

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Lambert, A. The genealogy of continuous-state branching processes with immigration. Probab Theory Relat Fields 122, 42–70 (2002). https://doi.org/10.1007/s004400100155

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  • Issue Date: January 2002

  • DOI: https://doi.org/10.1007/s004400100155

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Keywords

  • Recent Work
  • Time Process
  • Gall
  • Markov Process
  • Local Time
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