Maximal displacement in the $d$-dimensional branching Brownian motion

Bastien Mallein (1, 2)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

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

Mots clés: Brownian motion; branching process

Domaines: Probabilités [math.PR]

Bastien Mallein. Maximal displacement in the $d$-dimensional branching Brownian motion. Electronic Communications in Probability, 2015, ⟨10.1214/ECP.v20-4216⟩. ⟨hal-01322443⟩