## Brunet-derrida particle systems, free boundary problems and wiener-hopf equations

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 c ≥ a or c>a,wherea 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. © 2011 Institute of Mathematical Statistics.

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- Statistics & Probability
- 4905 Statistics
- 4904 Pure mathematics
- 0104 Statistics
- 0101 Pure Mathematics

### Citation

*Annals of Probability*,

*39*(6), 2043–2078. https://doi.org/10.1214/10-AOP601

*Annals of Probability*39, no. 6 (January 1, 2011): 2043–78. https://doi.org/10.1214/10-AOP601.

*Annals of Probability*, vol. 39, no. 6, Jan. 2011, pp. 2043–78.

*Scopus*, doi:10.1214/10-AOP601.

## Published In

## DOI

## ISSN

## Publication Date

## Volume

## Issue

## Start / End Page

## Related Subject Headings

- Statistics & Probability
- 4905 Statistics
- 4904 Pure mathematics
- 0104 Statistics
- 0101 Pure Mathematics