Selection principle for the $N$-BBM

TL;DR

This paper proves that as the number of particles in the $N$-BBM system grows large, its stationary distribution converges to a specific wave solution of a related PDE, resolving a long-standing open question.

Contribution

It establishes a new selection principle linking the $N$-BBM particle system to the minimal travelling wave of a PDE as $N$ approaches infinity.

Findings

Convergence of the empirical measure to the minimal travelling wave.

Resolution of an open question from prior literature.

Connection to recent results on Fleming-Viot systems.

Abstract

The $N$-branching Brownian motion with selection ($N$-BBM) is a particle system consisting of $N$ independent particles that diffuse as Brownian motions in $\mathbb{R}$, branch at rate one, and whose size is kept constant by removing the leftmost particle at each branching event. We establish the following selection principle: as $N \rightarrow \infty$ the stationary empirical measure of the $N$-particle system converges to the minimal travelling wave of the associated free boundary PDE. This resolves an open question going back at least to \cite[p.19]{Maillard2012} and \cite{GroismanJonckheer}, and follows a recent related result by the second author establishing a similar selection principle for the so-called Fleming-Viot particle system \cite{Tough23}.