November 2022 Brownian bees in the infinite swarm limit
Julien Berestycki, Éric Brunet, James Nolen, Sarah Penington
Author Affiliations +
Ann. Probab. 50(6): 2133-2177 (November 2022). DOI: 10.1214/22-AOP1578


The Brownian bees model is a branching particle system with spatial selection. It is a system of N particles which move as independent Brownian motions in Rd and independently branch at rate 1, and, crucially, at each branching event, the particle which is the furthest away from the origin is removed to keep the population size constant. In the present work we prove that, as N, the behaviour of the particle system is well approximated by the solution of a free boundary problem (which is the subject of a companion paper (Trans. Amer. Math. Soc. 374 (2021) 6269–6329)), the hydrodynamic limit of the system. We then show that for this model the so-called selection principle holds; that is, that as N, the equilibrium density of the particle system converges to the steady-state solution of the free boundary problem.

Funding Statement

The work of JN was partially funded through grant DMS-1351653 from the U.S. National Science Foundation.


The authors wish to thank Louigi Addario-Berry, Erin Beckman, Nathanaël Berestycki and Pascal Maillard for stimulating discussions at various points of this project.


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Julien Berestycki. Éric Brunet. James Nolen. Sarah Penington. "Brownian bees in the infinite swarm limit." Ann. Probab. 50 (6) 2133 - 2177, November 2022.


Received: 1 April 2021; Revised: 1 January 2022; Published: November 2022
First available in Project Euclid: 23 October 2022

MathSciNet: MR4499276
zbMATH: 1500.60056
Digital Object Identifier: 10.1214/22-AOP1578

Primary: 35R35 , 60J80
Secondary: 60K35 , 82C22

Keywords: branching processes , free boundary problems , interacting particle systems

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 6 • November 2022
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