Branching Brownian motion with an inhomogeneous breeding potential

This article concerns branching Brownian motion (BBM) with dyadic branching at rate β|y|p for a particle with spatial position y∈ℝ, where β>0. It is known that for p>2 the number of particles blows up almost surely in finite time, while for p=2 the expected number of particles alive blows up in finite time, although the number of particles alive remains finite almost surely, for all time. We define the right-most particle, Rt, to be the supremum of the spatial positions of the particles alive at time t and study the asymptotics of Rt as t→∞. In the case of constant breeding at rate β the linear asymptotic for Rt is long established. Here, we find asymptotic results for Rt in the case p∈(0, 2]. In contrast to the linear asymptotic in standard BBM we find polynomial asymptotics of arbitrarily high order as p↑2, and a non-trivial limit for lnRt when p=2. Our proofs rest on the analysis of certain additive martingales, and related spine changes of measure.