Maximal displacement of d-dimensional branching Brownian motion

Abstract

We consider a branching Brownian motion evolving in ℝd. 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 ≥ 2, individuals on the frontier of the process are close parents if and only if they are geographically close.