The almost-sure population growth rate in branching Brownian motion with a quadratic breeding potential

In this note we consider a branching Brownian motion (BBM) on $\mathbb{R}$ in which a particle at spatial position $y$ splits into two at rate $βy^2$, where $β>0$ is a constant. This is a critical breeding rate for BBM in the sense that the expected population size blows up in finite time while the population size remains finite, almost surely, for all time. We find an asymptotic for the almost sure rate of growth of the population.