Electronic Journal of Probability

An ergodic theorem for the frontier of branching Brownian motion

Louis-Pierre Arguin, Anton Bovier, and Nicola Kistler

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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, 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 [Comm. Pure Appl. Math. 64 (2011)].

Article information

Electron. J. Probab., Volume 18 (2013), paper no. 53, 25 pp.

Accepted: 13 May 2013
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G70: Extreme value theory; extremal processes 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Branching Brownian motion ergodicity extreme value theory KPP equation and traveling waves

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Arguin, Louis-Pierre; Bovier, Anton; Kistler, Nicola. An ergodic theorem for the frontier of branching Brownian motion. Electron. J. Probab. 18 (2013), paper no. 53, 25 pp. doi:10.1214/EJP.v18-2082. https://projecteuclid.org/euclid.ejp/1465064278

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