THE GENEALOGY OF BRANCHING BROWNIAN MOTION WITH ABSORPTION

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...

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Bibliographic Details
Published in:The Annals of probability 2013-03, Vol.41 (2), p.527-618
Main Authors: Berestycki, Julien, Berestycki, Nathanaël, Schweinsberg, Jason
Format: Article
Language:English
Subjects:
60F17
60G15
60J80
60J99
Atoms & subatomic particles
Bolthausen–Sznitman coalescent
Branching Brownian motion
Brownian motion
continuous-state branching processes
Descendants
Genealogy
Mathematical constants
Mathematics
Particle motion
Perceptron convergence procedure
Population genetics
Population size
Probability
Random variables
Random walk
Stochastic models
Studies
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Description
Summary: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.
ISSN:0091-1798
2168-894X
DOI:10.1214/11-AOP728