The Annals of Probability

Infinitely ramified point measures and branching Lévy processes

Jean Bertoin and Bastien Mallein

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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.

Article information

Source
Ann. Probab., Volume 47, Number 3 (2019), 1619-1652.

Dates
Received: March 2017
Revised: January 2018
First available in Project Euclid: 2 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.aop/1556784028

Digital Object Identifier
doi:10.1214/18-AOP1292

Mathematical Reviews number (MathSciNet)
MR3945755

Zentralblatt MATH identifier
07067278

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60G51: Processes with independent increments; Lévy processes 60G55: Point processes

Keywords
Branching random walk Lévy process growth-fragmentation infinitely ramified point measure

Citation

Bertoin, Jean; Mallein, Bastien. Infinitely ramified point measures and branching Lévy processes. Ann. Probab. 47 (2019), no. 3, 1619--1652. doi:10.1214/18-AOP1292. https://projecteuclid.org/euclid.aop/1556784028


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