We prove a weak limit theorem which relates the large time behavior of the maximum of a branching Brownian motion to the limiting value of a certain associated martingale. This exhibits the minimal velocity travelling wave for the KPP-Fisher equation as a translation mixture of extreme-value distributions. We also show that every particle in a branching Brownian motion has a descendant at the frontier at some time. A final section states several conjectures concerning a hypothesized stationary "standing wave of particles" process and the relationship of this process to branching Brownian motion.
S. P. Lalley. T. Sellke. "A Conditional Limit Theorem for the Frontier of a Branching Brownian Motion." Ann. Probab. 15 (3) 1052 - 1061, July, 1987. https://doi.org/10.1214/aop/1176992080