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The genealogy of branching Brownian motion with absorption. (English) Zbl 1304.60088

The purpose of the paper is to provide rigorous versions of some conjectures which come from the works of E. Brunet et al. [“Noisy traveling waves: effect of selection on genealogies ”, Europhys. Lett. 76, No. 1, 1–7 (2006; doi:10.1209/epl/i2006-10224-4); “Effect of selection on ancestry: an exactly soluble case and its phenomenological generalization”, Phys. Rev. E (3) 76, No. 4, 041104, 20 p. (2007; doi:10.1103/PhysRevE.76.041104)] and concerning the effect of natural selection on the genealogy of a population.
In the present paper, the authors 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. They focus on understanding the genealogy of a sample from the population after a large time. It is shown that the time to the most recent common ancestor of a sample behaves like \((\log)^3\).
Moreover, the authors identify the limiting geometry of the coalescence tree of a sample, which (as shown in the paper) is governed by a coalescent process known as the Bolthausen-Sznitman coalescent.

MSC:

60J65 Brownian motion
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60F17 Functional limit theorems; invariance principles
60G15 Gaussian processes
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References:

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