Survival of Near-Critical Branching Brownian Motion

@article{Berestycki2011SurvivalON,
  title={Survival of Near-Critical Branching Brownian Motion},
  author={Julien Berestycki and Nathanael Berestycki and Jason Schweinsberg},
  journal={Journal of Statistical Physics},
  year={2011},
  volume={143},
  pages={833-854}
}
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 Qμ(x). It is proved that if $L=\pi/\sqrt{\varepsilon}$ then for all x∈ℝ, lim ε→0Qμ(L+x)=θ(x)∈(0,1) exists and… 

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