Survival probabilities for branching Brownian motion with absorption (original) (raw)

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 x (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 ∈ F s .