Survival of Near-Critical Branching Brownian Motion |
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Authors: | Julien Berestycki Nathanaël Berestycki Jason Schweinsberg |
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Affiliation: | 1.Université Pierre et Marie Curie (Paris VI)—CNRS,Paris,France;2.University of Cambridge,Cambridge,UK;3.University of California at San Diego,La Jolla,USA |
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Abstract: | Consider a system of particles performing branching Brownian motion with negative drift (mu= sqrt{2 - varepsilon}) and killed upon hitting zero. Initially there is one particle at x>0. Kesten (Stoch. Process. Appl. 7:9–47, 1978) showed that the process survives with positive probability if and only if ε>0. Here we are interested in the asymptotics as ε→0 of the survival probability Q μ (x). It is proved that if (L=pi/sqrt{varepsilon}) then for all x∈?, lim? ε→0 Q μ (L+x)=θ(x)∈(0,1) exists and is a traveling 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 (Berestycki et al. in arXiv:1001.2337, 2010). This completes earlier work by Harris, Harris and Kyprianou (Ann. Inst. Henri Poincaré Probab. Stat. 42:125–145, 2006) and confirms predictions made by Derrida and Simon (Europhys. Lett. 78:60006, 2007), which were obtained using nonrigorous PDE methods. |
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