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

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

2009-07-29

arxiv.org/abs/0907.5180v4

Annals of Probability 2011, Vol. 39, No. 6, 2043-2078

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/10-AOP601 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/10-AOP601

We consider a branching-selection system in $\mathbb {R}$ 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 $N\to\infty$, 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 $c\geq a$ or $c

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