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

Mallein, Bastien

Electron. Commun. Probab. 20, Paper No. 76, 12 p. (2015).

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.)

Keywords:

branching Brownian motion; maximal displacement