2010/09/02 by Julien Berestycki, Nathanaël Berestycki, Jason Schweinsberg
Mathematics · Physics and Astronomy · #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR #msc:60F17 #msc:60G15 #msc:60J80 #msc:60J99 #stochastic dynamics and bifurcation
paper · pdf · doi:10.1007/s10955-011-0224-9
arxiv created 2010/09/02 · openalex publication_date 2011/05/27 · arxiv updated 2015/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider a system of particles performing branching Brownian motion with negative drift μ= √(2 - ε) and killed upon hitting zero. Initially there is one particle at x>0. Kesten showed that the process survives with positive probability if and only if ε>0. Here we are interested in the asymptotics as \eps→ 0 of the survival probability Qμ(x). It is proved that if L= π/√ε then for all x ∈ \R, limε→ 0 Qμ(L+x) = θ(x) ∈ (0,1) exists and is a travelling wave solution of the Fisher-KPP equation. Furthermore, we obtain sharp asymptotics of the survival probability when x<L and L-x → ∞. The proofs rely on probabilistic methods developed by the authors in a previous work. This completes earlier work by Harris, Harris and Kyprianou and confirms predictions made by Derrida and Simon, which were obtained using nonrigorous PDE methods.