Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

An ergodic theorem for the extremal process of branching Brownian motion

Louis-Pierre Arguin, Anton Bovier, and Nicola Kistler

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Abstract

In a previous paper, the authors proved a conjecture of Lalley and Sellke that the empirical (time-averaged) distribution function of the maximum of branching Brownian motion converges almost surely to a Gumbel distribution. The result is extended here to the entire system of particles that are extremal, i.e. close to the maximum. Namely, it is proved that the distribution of extremal particles under time-average converges to a Poisson cluster process.

Résumé

Dans un article précédent, les auteurs ont démontré une conjecture de Lalley et Sellke stipulant que la loi empirique (en faisant la moyenne sur les temps) du maximum du mouvement brownien branchant converge presque sûrement vers une loi de Gumbel. Ce résultat est généralisé ici au système de particules extrémales, c’est-à-dire celles se situant près du maximum. Précisément, il est démontré que la loi conjointe empirique des positions des particules extrémales converge vers la loi d’un processus poissonien de nuages.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 51, Number 2 (2015), 557-569.

Dates
First available in Project Euclid: 10 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1428672682

Digital Object Identifier
doi:10.1214/14-AIHP608

Mathematical Reviews number (MathSciNet)
MR3335016

Zentralblatt MATH identifier
1315.60063

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

Keywords
Branching Brownian motion Ergodicity Extreme value theory KPP equation and traveling waves

Citation

Arguin, Louis-Pierre; Bovier, Anton; Kistler, Nicola. An ergodic theorem for the extremal process of branching Brownian motion. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 2, 557--569. doi:10.1214/14-AIHP608. https://projecteuclid.org/euclid.aihp/1428672682


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References

  • [1] E. Aïdekon, J. Berestycki, E. Brunet and Z. Shi. The branching Brownian motion seen from its tip. Probab. Theory Related Fields 157 (2013) 405–451.
  • [2] L.-P. Arguin, A. Bovier and N. Kistler. The genealogy of extremal particles of branching Brownian motion. Comm. Pure Appl. Math. 64 (2011) 1647–1676.
  • [3] L.-P. Arguin, A. Bovier and N. Kistler. Poissonian statistics in the extremal process of branching Brownian motion. Ann. Appl. Probab. 22 (2012) 1693–1711.
  • [4] L.-P. Arguin, A. Bovier and N. Kistler. The extremal process of branching Brownian motion. Probab. Theory Related Fields 157 (2013) 535–574.
  • [5] L.-P. Arguin, A. Bovier and N. Kistler. An ergodic theorem for the frontier of branching Brownian motion. Electron. J. Probab. 18 (53) (2013) 1–25.
  • [6] M. Bramson. Maximal displacement of branching Brownian motion. Comm. Pure Appl. Math. 31 (1978) 531–581.
  • [7] M. Bramson. Convergence of solutions of the Kolmogorov equation to travelling waves. Mem. Amer. Math. Soc. 44 (285) (1983) iv+190.
  • [8] E. Brunet and B. Derrida. Statistics at the tip of a branching random walk and the delay of travelling waves. Euro Phys. Lett. 87 (2009) 60010.
  • [9] E. Brunet and B. Derrida. A branching random walk seen from the tip. J. Stat. Phys. 143 (2010) 420–446.
  • [10] B. Chauvin and A. Rouault. Supercritical branching Brownian motion and K–P–P equation in the critical speed-area. Math. Nachr. 149 (1990) 41–59.
  • [11] R. A. Fisher. The wave of advance of advantageous genes. Ann. Eugen. 7 (1937) 355–369.
  • [12] J.-B. Gouéré. Branching Brownian motion seen from its left-most particle. Séminaire Bourbaki, 65ème année, no. 1067, 2013. Available at arXiv:1305.4396.
  • [13] O. Kallenberg. Random Measures. Springer, Berlin, 1986.
  • [14] A. Kolmogorov, I. Petrovsky and N. Piscounov. Etude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Bull. Univ. Moskov Ser. Internat. Sect. 1 (1937) 1–25.
  • [15] S. P. Lalley and T. Sellke. A conditional limit theorem for the frontier of a branching Brownian motion. Ann. Probab. 15 (1987) 1052–1061.
  • [16] R. Lyons. Strong laws of large numbers for weakly correlated random variables. Michigan Math. J. 35 (1988) 353–359.
  • [17] H. P. McKean. Application of Brownian motion to the equation of Kolmogorov–Petrovskii–Piskunov. Comm. Pure Appl. Math. 28 (1976) 323–331.
  • [18] J. Neveu. Processus ponctuels. In École d’Été de Probabilités de Saint-Flour VI – 1976 249–445. Lecture Notes in Math. 598. Springer, Berlin, 1977.
  • [19] S. Sawyer. Branching diffusion processes in population genetics. Adv. in Appl. Probab. 8 (1976) 659–689.