The Annals of Probability

The genealogy of branching Brownian motion with absorption

Julien Berestycki, Nathanaël Berestycki, and Jason Schweinsberg

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Abstract

We consider a system of particles which perform branching Brownian motion with negative drift and are killed upon reaching zero, in the near-critical regime where the total population stays roughly constant with approximately $N$ particles. We show that the characteristic time scale for the evolution of this population is of order $(\log N)^{3}$, in the sense that when time is measured in these units, the scaled number of particles converges to a variant of Neveu’s continuous-state branching process. Furthermore, the genealogy of the particles is then governed by a coalescent process known as the Bolthausen–Sznitman coalescent. This validates the nonrigorous predictions by Brunet, Derrida, Muller and Munier for a closely related model.

Article information

Source
Ann. Probab., Volume 41, Number 2 (2013), 527-618.

Dates
First available in Project Euclid: 8 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1362750935

Digital Object Identifier
doi:10.1214/11-AOP728

Mathematical Reviews number (MathSciNet)
MR3077519

Zentralblatt MATH identifier
1304.60088

Subjects
Primary: 60J99: None of the above, but in this section
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F17: Functional limit theorems; invariance principles 60G15: Gaussian processes

Keywords
Branching Brownian motion Bolthausen–Sznitman coalescent continuous-state branching processes

Citation

Berestycki, Julien; Berestycki, Nathanaël; Schweinsberg, Jason. The genealogy of branching Brownian motion with absorption. Ann. Probab. 41 (2013), no. 2, 527--618. doi:10.1214/11-AOP728. https://projecteuclid.org/euclid.aop/1362750935


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