The Annals of Probability

Infinitely ramified point measures and branching Lévy processes

Jean Bertoin and Bastien Mallein

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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