On the density of branching Brownian motion

Authors : Mehmet OZ

Pages : 229-247

Doi:10.15672/hujms.1016517

View : 12 | Download : 5

Publication Date : 2023-02-15

Article Type : Research Paper

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 $rinsert ignore into journalissuearticles values(t);$-enlargement of the support of branching Brownian motion when $rinsert ignore into journalissuearticles values(t);$ 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.
Keywords : Branching Brownian motion, density, large deviations, neighborhood recurrence