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.
Publié le : 1998-01-14
Classification:
Branching processes,
Lévy processes,
genealogy,
local time,
exploration process,
random tree,
jump processes,
60J80,
60J30,
60J25,
60J55
@article{1022855417,
author = {Le Gall, Jean-Francois and Le Jan, Yves},
title = {Branching processes in L\'evy processes: the exploration
process},
journal = {Ann. Probab.},
volume = {26},
number = {1},
year = {1998},
pages = { 213-252},
language = {en},
url = {http://dml.mathdoc.fr/item/1022855417}
}
Le Gall, Jean-Francois; Le Jan, Yves. Branching processes in Lévy processes: the exploration
process. Ann. Probab., Tome 26 (1998) no. 1, pp. 213-252. http://gdmltest.u-ga.fr/item/1022855417/