The genealogy of branching Brownian motion with absorption

TL;DR

This paper analyzes a near-critical branching Brownian motion with absorption, revealing that its population evolution occurs on a $( ext{log } N)^3$ timescale and its genealogy follows the Bolthausen-Sznitman coalescent, confirming prior predictions.

Contribution

It establishes the $( ext{log } N)^3$ timescale for population evolution and identifies the genealogy as the Bolthausen-Sznitman coalescent in a near-critical branching Brownian motion with absorption.

Findings

Population size remains roughly constant over $( ext{log } N)^3$ time scale.

Scaled population converges to Neveu's continuous-state branching process.

Genealogy follows the Bolthausen-Sznitman coalescent.

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.