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We consider one-dimensional branching Brownian motion in which particles areabsorbed at the origin. We assume that when a particle branches, the offspringdistribution is supercritical, but the particles are given a critical drifttowards the origin so that the process eventually goes extinct with probabilityone. We establish precise asymptotics for the probability that the processsurvives for a large time t, building on previous results by Kesten (1978) andBerestycki, Berestycki, and Schweinsberg (2014). We also prove a Yaglom-typelimit theorem for the behavior of the process conditioned to survive for anunusually long time, providing an essentially complete answer to a questionfirst addressed by Kesten (1978). An important tool in the proofs of theseresults is the convergence of a certain observable to a continuous statebranching process. Our proofs incorporate new ideas which might be of use inother branching models.

Yaglom-type limit theorems for branching Brownian motion with absorption