Limiting distributions for the maximal displacement of branching Brownian motions

Abstract

We determine the long time behavior and the exact order of the tail probability for the maximal displacement of a branching Brownian motion in Euclidean space in terms of the principal eigenvalue of the associated Schr\"odinger type operator. To establish our results, we show a sharp and locally uniform growth order of the Feynman-Kac semigroup.