Maximal displacement in the $$d$$-dimensional branching Brownian motion.(English)Zbl 1329.60307

Summary: We consider a branching Brownian motion evolving in $$\mathbb{R}^d$$. We prove that the asymptotic behaviour of the maximal displacement is given by a first ballistic order, plus a logarithmic correction that increases with the dimension $$d$$. The proof is based on simple geometrical evidence. It leads to the interesting following side result: with high probability, for any $$d \geq 2$$, individuals on the frontier of the process are close parents if and only if they are geographically close.

MSC:

 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60J65 Brownian motion 60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
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