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We study a branching Brownian motion ZZ in RdRd, 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 ZZ hits a trap, asymptotically in time tt. 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

Conditional speed of branching Brownian motion, skeleton decomposition and application to random obstacles