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November 2011 Brunet–Derrida particle systems, free boundary problems and Wiener–Hopf equations
Rick Durrett, Daniel Remenik
Ann. Probab. 39(6): 2043-2078 (November 2011). DOI: 10.1214/10-AOP601


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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Rick Durrett. Daniel Remenik. "Brunet–Derrida particle systems, free boundary problems and Wiener–Hopf equations." Ann. Probab. 39 (6) 2043 - 2078, November 2011.


Published: November 2011
First available in Project Euclid: 17 November 2011

zbMATH: 1243.60066
MathSciNet: MR2932664
Digital Object Identifier: 10.1214/10-AOP601

Primary: 35C07 , 35R35 , 60F99 , 60J80 , 60J99

Keywords: Branching random walk , Branching-selection system , free boundary equation , traveling wave solutions , Wiener–Hopf equation

Rights: Copyright © 2011 Institute of Mathematical Statistics


Vol.39 • No. 6 • November 2011
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