Conditional Speed of BBM, Skeleton Decomposition and Application to Random Obstacles

We study a branching Brownian motion $$Z$$ in $$\mathbb{R}^d$$, among obstacles scattered according to a Poisson random measure with a radially decaying intensity. Obstacles are balls with constant radius and each one works as a trap for the whole motion when hit by a particle. Considering a general offspring distribution, we derive the decay rate of the annealed probability that none of the particles of $$Z$$ hits a trap, asymptotically in time $$t$$. This proves to be a rich problem motivating the proof of a more general result about the speed of branching Brownian motion conditioned on non-extinction. We provide an appropriate "skeleton" decomposition for the underlying Galton-Watson process when supercritical and show that the "doomed" particles do not contribute to the asymptotic decay rate.

This is joint work with M. Caglar and M. Oz (Istanbul); to appear in AIHP.