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May 2019 Infinitely ramified point measures and branching Lévy processes
Jean Bertoin, Bastien Mallein
Ann. Probab. 47(3): 1619-1652 (May 2019). DOI: 10.1214/18-AOP1292

Abstract

We call a random point measure infinitely ramified if for every $n\in\mathbb{N}$, it has the same distribution as the $n$th generation of some branching random walk. On the other hand, branching Lévy processes model the evolution of a population in continuous time, such that individuals move in space independently, according to some Lévy process, and further beget progenies according to some Poissonian dynamics, possibly on an everywhere dense set of times. Our main result connects these two classes of processes much in the same way as in the case of infinitely divisible distributions and Lévy processes: the value at time $1$ of a branching Lévy process is an infinitely ramified point measure, and conversely, any infinitely ramified point measure can be obtained as the value at time $1$ of some branching Lévy process.

Citation

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Jean Bertoin. Bastien Mallein. "Infinitely ramified point measures and branching Lévy processes." Ann. Probab. 47 (3) 1619 - 1652, May 2019. https://doi.org/10.1214/18-AOP1292

Information

Received: 1 March 2017; Revised: 1 January 2018; Published: May 2019
First available in Project Euclid: 2 May 2019

zbMATH: 07067278
MathSciNet: MR3945755
Digital Object Identifier: 10.1214/18-AOP1292

Subjects:
Primary: 60G51, 60G55, 60J80

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 3 • May 2019
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