The Annals of Probability

Brunet–Derrida particle systems, free boundary problems and Wiener–Hopf equations

Rick Durrett and Daniel Remenik

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We consider a branching-selection system in ℝ with N particles which give birth independently at rate 1 and where after each birth the leftmost particle is erased, keeping the number of particles constant. We show that, as N → ∞, the empirical measure process associated to the system converges in distribution to a deterministic measure-valued process whose densities solve a free boundary integro-differential equation. We also show that this equation has a unique traveling wave solution traveling at speed c or no such solution depending on whether ca or c < a, where a is the asymptotic speed of the branching random walk obtained by ignoring the removal of the leftmost particles in our process. The traveling wave solutions correspond to solutions of Wiener–Hopf equations.

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Ann. Probab., Volume 39, Number 6 (2011), 2043-2078.

First available in Project Euclid: 17 November 2011

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Primary: 60J99: None of the above, but in this section 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 35R35: Free boundary problems 35C07: Traveling wave solutions 60F99: None of the above, but in this section

Branching-selection system branching random walk free boundary equation Wiener–Hopf equation traveling wave solutions


Durrett, Rick; Remenik, Daniel. Brunet–Derrida particle systems, free boundary problems and Wiener–Hopf equations. Ann. Probab. 39 (2011), no. 6, 2043--2078. doi:10.1214/10-AOP601.

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