The Annals of Probability

Branching processes in Lévy processes: the exploration process

Jean-Francois Le Gall and Yves Le Jan

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The main idea of the present work is to associate with a general continuous branching process an exploration process that contains the desirable information about the genealogical structure. The exploration process appears as a simple local time functional of a Lévy process with no negative jumps, whose Laplace exponent coincides with the branching mechanism function. This new relation between spectrally positive Lévy processes and continuous branching processes provides a unified perspective on both theories. In particular, we derive the adequate formulation of the classical Ray–Knight theorem for such Lévy processes. As a consequence of this theorem, we show that the path continuity of the exploration process is equivalent to the almost sure extinction of the branching process.

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Ann. Probab., Volume 26, Number 1 (1998), 213-252.

First available in Project Euclid: 31 May 2002

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Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J30
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60J55: Local time and additive functionals

Branching processes Lévy processes genealogy local time exploration process random tree jump processes


Le Gall, Jean-Francois; Le Jan, Yves. Branching processes in Lévy processes: the exploration process. Ann. Probab. 26 (1998), no. 1, 213--252. doi:10.1214/aop/1022855417.

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