, Volume 157, Issue 3-4, pp 535-574

The extremal process of branching Brownian motion

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Abstract

We prove that the extremal process of branching Brownian motion, in the limit of large times, converges weakly to a cluster point process. The limiting process is a (randomly shifted) Poisson cluster process, where the positions of the clusters is a Poisson process with intensity measure with exponential density. The law of the individual clusters is characterized as branching Brownian motions conditioned to perform “unusually large displacements”, and its existence is proved. The proof combines three main ingredients. First, the results of Bramson on the convergence of solutions of the Kolmogorov–Petrovsky–Piscounov equation with general initial conditions to standing waves. Second, the integral representations of such waves as first obtained by Lalley and Sellke in the case of Heaviside initial conditions. Third, a proper identification of the tail of the extremal process with an auxiliary process (based on the work of Chauvin and Rouault), which fully captures the large time asymptotics of the extremal process. The analysis through the auxiliary process is a rigorous formulation of the cavity method developed in the study of mean field spin glasses.

L.-P. Arguin was supported by the NSF grant DMS-0604869 during this work. L.-P. Arguin also acknowledges support from NSERC Canada and FQRNT Québec. A. Bovier is partially supported through the German Research Council in the SFB 611 and the Hausdorff Center for Mathematics. N. Kistler is partially supported by the Hausdorff Center for Mathematics. Part of this work has been carried out during the Junior Trimester Program Stochastics at the Hausdorff Center in Bonn, and during a visit by the third named author to the Courant Institute at NYU: hospitality and financial support of both institutions are gratefully acknowledged.