Survival of near-critical branching Brownian motion
Résumé
Consider a system of particles performing branching Brownian motion with negative drift mu = root 2-epsilon 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 e > 0. Here we are interested in the asymptotics as e. 0 of the survival probability Q(mu)(x). It is proved that if L = p/v e then for all x. R, lim(epsilon -> 0) Q(mu)(L + x) = theta(x) is an element of (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 -> infinity. 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 Poincare 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.