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