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An ergodic theorem for the frontier of branching Brownian motion. (English) Zbl 1286.60082

Summary: We prove a conjecture of S. P. Lalley and T. Sellke [Ann. Probab. 15, 1052–1061 (1987; Zbl 0622.60085)] asserting that the empirical (time-averaged) distribution function of the maximum of branching Brownian motion converges almost surely to a double exponential, or Gumbel, distribtion 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 [Commun. Pure Appl. Math. 64, No. 12, 1647–1676 (2011; Zbl 1236.60081)].

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60G70 Extreme value theory; extremal stochastic processes
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
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