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A conditional limit theorem for the frontier of a branching Brownian motion. (English) Zbl 0622.60085
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.

60J60 Diffusion processes
60F05 Central limit and other weak theorems
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