Population stabilization in branching brownian motion with absorption and drift

We consider, through PDE methods, branching Brownian motion with drift and absorption. It is well known that there exists a critical drift which separates those processes which die out almost surely from those which survive with positive probability. In this work, we consider lower-order corrections to the critical drift which ensures a nonnegative, bounded expected number of particles and convergence of this expectation to a limiting nonnegative number which is positive for some initial data. In particular, we show that the average number of particles stabilizes at the convergence rate O(log(t)/t) if and only if the multiplicative factor of the O(t^-1/2) correction term is 3√πt^-1/2. Otherwise, the convergence rate is O(1/√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.