Tue, 16 Jan 2024
16:00 - 17:00
Julien Berestycki
Department of Statistics, University of Oxford
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. It is a very simple model for the evolution of a population under selection that has generated some fascinating research since its introduction by Brunet and Derrida in the early 2000s.
If one recentre the positions by the position of the left most particle, this system has a stationary distribution. I will show that, as $N\to\infty$ the stationary empirical measure of the $N$-particle system converges to the minimal travelling wave of an associated free boundary PDE. This resolves an open question going back at least to works of e.g. Maillard in 2012.
It follows a recent related result by Oliver Tough (with whom this is joint work) establishing a similar selection principle for the so-called Fleming-Viot particle system.
With very best wishes,
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