An ergodic theorem for the frontier of branching Brownian motion

Details An ergodic theorem for the frontier of branching Brownian motion An ergodic theorem for the frontier of branching Brownian motion

2012-01-09

arxiv.org/abs/1201.1701v1

Mathematics

Probability

4 figures

Scientific paper

We prove a conjecture of Lalley and Sellke [Ann. Probab. 15 (1987)] asserting that the empirical (time-averaged) distribution function of the maximum of branching Brownian motion converges almost surely to a double exponential, or Gumbel, distribution with a random shift. The method of proof is based on the decorrelation of the maximal displacements for appropriate time scales. A crucial input is the localization of the paths of particles close to the maximum that was previously established by the authors [Comm. Pure Appl. Math. 64 (2011)].

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