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arXiv · 1001.2337

The genealogy of branching Brownian motion with absorption

Julien Berestycki·Nathanaël Berestycki·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.

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Julien Berestycki, Nathanaël Berestycki, Jason Schweinsberg. 2010-01-13. The genealogy of branching Brownian motion with absorption. https://doi.org/10.1214/11-aop728

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