Mathematics – Probability
Scientific paper
2010-10-12
Mathematics
Probability
16 pages, 2 figures, Final version accepted for publication
Scientific paper
As a first step towards a characterization of the limiting extremal process of branching Brownian motion, we proved in a recent work that, in the limit of large time $t$, extremal particles descend with overwhelming probability from ancestors having split either within a distance of order one from time 0, or within a distance of order one from time $t$. The result suggests that the extremal process of branching Brownian motion is a randomly shifted cluster point process. Here we put part of this picture on rigorous ground: we prove that the point process obtained by retaining only those extremal particles which are also maximal inside the clusters converges in the limit of large $t$ to a random shift of a Poisson point process with exponential density. The last section discusses the Tidal Wave Conjecture by Lalley and Sellke on the full limiting extremal process and its relation to the work of Chauvin and Rouault on branching Brownian motion with atypical displacement.
Arguin Louis-Pierre
Bovier Anton
Kistler Nicola
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