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We consider branching Brownian motion in which initially there is oneparticle at $x$, particles produce a random number of offspring with mean $m+1$at the time of branching events, and each particle branches at rate $\beta =1/2m$. Particles independently move according to Brownian motion with drift$-1$ and are killed at the origin. It is well-known that this processeventually dies out with positive probability. We condition this process tosurvive for an unusually large time $t$ and study the behavior of the processat small times $s \ll t$ using a spine decomposition. We show, in particular,that the time when a particle gets furthest from the origin is of the order$t^{5/6}$.

The all-time maximum for branching Brownian motion with absorption conditioned on long-time survival