An ergodic theorem for the extremal process of branching brownian motion
Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 2, p. 557-569

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.

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.

DOI : https://doi.org/10.1214/14-AIHP608
Classification:  60J80,  60G70,  82B44
Keywords: branching brownian motion, ergodicity, extreme value theory, KPP equation and traveling waves
@article{AIHPB_2015__51_2_557_0,
     author = {Arguin, Louis-Pierre and Bovier, Anton and Kistler, Nicola},
     title = {An ergodic theorem for the extremal process of branching brownian motion},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {51},
     number = {2},
     year = {2015},
     pages = {557-569},
     doi = {10.1214/14-AIHP608},
     zbl = {1286.60045},
     mrnumber = {3335016},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2015__51_2_557_0}
}
Arguin, Louis-Pierre; Bovier, Anton; Kistler, Nicola. An ergodic theorem for the extremal process of branching brownian motion. Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 2, pp. 557-569. doi : 10.1214/14-AIHP608. http://www.numdam.org/item/AIHPB_2015__51_2_557_0/

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