The genealogy of branching Brownian motion with absorption

Details The genealogy of branching Brownian motion with absorption The genealogy of branching Brownian motion with absorption

2010-01-13

arxiv.org/abs/1001.2337v2

Mathematics

Probability

78 pages, a few typos corrected. To appear in Annals of Probability

Scientific paper

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 non-rigorous predictions by Brunet, Derrida, Muller, and Munier for a closely related model.

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