Survival probabilities for branching Brownian motion with absorption

Abstract

We study a branching Brownian motion (BBM) with absorption, in which particles move as Brownian motions with drift $-\rho$, undergo dyadic branching at rate $\beta>0$, and are killed on hitting the origin. In the case $\rho>\sqrt{2\beta}$ the extinction time for this process, $\zeta$, is known to be finite almost surely. The main result of this article is a large-time asymptotic formula for the survival probability $P^x(\zeta>t)$ in the case $\rho>\sqrt{2\beta}$, where $P^x$ is the law of the BBM with absorption started from a single particle at the position $x>0$. We also introduce an additive martingale, $V$, for the BBM with absorption, and then ascertain the convergence properties of $V$. Finally, we use $V$ in a `spine' change of measure and interpret this in terms of `conditioning the BBM to survive forever' when $\rho>\sqrt{2\beta}$, in the sense that it is the large $t$-limit of the conditional probabilities $P^x(A\mid \zeta > t+s)$, for $A\in F_s$.