Branching Brownian motion in a strip: Survival near criticality

We consider a branching Brownian motion with linear drift in which particles are killed on exiting the interval (0,K) and study the evolution of the process on the event of survival as the width of the interval shrinks to the critical value at which survival is no longer possible. We combine spine techniques and a backbone decomposition to obtain exact asymptotics for the near-critical survival probability. This allows us to deduce the existence of a quasi-stationary limit result for the process conditioned on survival which reveals that the backbone thins down to a spine as we approach criticality. This paper is motivated by recent work on survival of near critical branching Brownian motion with absorption at the origin by Aïdékon and Harris [Near-critical survival probability of branching Brownian motion with an absorbing barrier (2010) Unpublished manuscript] as well as the work of Berestycki et al. [Ann. Probab. 41 (2013) 527.618; J. Stat. Phys. 143 (2011) 833.854].