Survival of Near-Critical Branching Brownian Motion

Consider a system of particles performing branching Brownian motion with negative drift, 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, then for all x∈ℝ, lim ε→0Qμ(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