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January 1998 Branching processes in Lévy processes: the exploration process
Jean-Francois Le Gall, Yves Le Jan
Ann. Probab. 26(1): 213-252 (January 1998). DOI: 10.1214/aop/1022855417


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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Jean-Francois Le Gall. Yves Le Jan. "Branching processes in Lévy processes: the exploration process." Ann. Probab. 26 (1) 213 - 252, January 1998.


Published: January 1998
First available in Project Euclid: 31 May 2002

zbMATH: 0948.60071
MathSciNet: MR1617047
Digital Object Identifier: 10.1214/aop/1022855417

Primary: 60J30 , 60J80
Secondary: 60J25 , 60J55

Keywords: branching processes , Exploration process , genealogy , Jump processes , Lévy processes , Local time , Random tree

Rights: Copyright © 1998 Institute of Mathematical Statistics

Vol.26 • No. 1 • January 1998
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