Population Stabilization in Branching Brownian Motion With Absorption

We consider, through PDE methods, branching Brownian motion with drift and absorption. It is well know that there exists a critical drift which separates those processes which die out almost surely and those which survive with positive probability. In this work, we consider lower order terms to the critical drift which ensures a non-negative, bounded expected number of particles and convergence of this expectation to a limiting number, $\alpha_0\geq 0$, which is positive for some initial data. In particular, we show that, in an average sense, the expected number of particles stabilizes to $\alpha_0$ like $O(\log(t)/t)$ if and only if the multiplicative factor of the $O(t^{-1/2})$ correction term is $3\sqrt{\pi} t^{-1/2}$. Otherwise, the convergence is like $O(1/\sqrt{t})$. We point out some connections between this work and recent work investigating the expansion of the front location for the initial value problem in Fisher-KPP.