Survival Probabilities for Branching Brow- Nian Motion with Absorption

We study a branching Brownian motion (BBM) with absorption, in which particles move as Brownian motions with drift −ρ, undergo dyadic branching at rate β > 0, and are killed on hitting the origin. In the case ρ > √ 2β the extinction time for this process, ζ, is known to be finite almost surely. The main result of this article is a large-time asymptotic formula for the survival probability P (ζ > t) in the case ρ > √ 2β, 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 ρ > √ 2β, in the sense that it is the large t-limit of the conditional probabilities P x(A|ζ > t+ s), for A ∈ Fs.