Mass of branching Brownian motion in an expanding ball

Abstract

Consider a 𝑑-dimensional branching Brownian motion starting with a single particle at the origin and let 𝑛_𝑡 be the number of particles at time 𝑡 whose ancestral lines have remained up to 𝑡 within a ball of radius 𝑟(𝑡) centered at the origin, where 𝑟(𝑡) increases sublinearly with 𝑡. We obtain a full limit large-deviation result as time tends to infinity on the probability that 𝑛_𝑡 is atypically small. A phase transition is identified, at which the nature of the optimal strategy to realize the aforementioned large-deviation event changes, and the Lyapunov exponent giving the decay rate of the associated large-deviation probability is continuous. As a corollary, we also obtain a kind of law of large numbers for 𝑛_𝑡 under the stronger assumption that 𝑟(𝑡) increases subdiffusively with 𝑡.