On the density of branching Brownian motion

Abstract

We consider a $d$-dimensional dyadic branching Brownian motion, and study the density of its support in the region where there is typically exponential growth of particles. Using geometric arguments and an extension of a previous result on the probability of absence of branching Brownian motion in linearly moving balls of fixed size, we obtain sharp asymptotic results on the covering radius of the support of branching Brownian motion, which is a measure of its density. As a corollary, we obtain large deviation estimates on the volume of the $r(t)$-enlargement of the support of branching Brownian motion when $r(t)$ decays exponentially in time $t$. As a by-product, we obtain the lower tail asymptotics for the mass of branching Brownian motion falling in linearly moving balls of exponentially shrinking radius, which is of independent interest.