The extremal process of branching Brownian motion

By L. Arguin, Anton Bovier, N. Kistler. Probability theory and related fields, 2011

194 citations recorded by the source corpus.

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.

Subfields: Condensed Matter Physics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/144278d1da5926f3bebfd180aa8f2acbd42e53c5

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Condensed Matter Physics

1. Zhan Shi
2. Thomas Madaule
3. David Belius
4. N. Kistler
5. L. Arguin
6. Elie Aidékon
7. Ofer Zeitouni
8. Eliran Subag

In Finance

1. Zhan Shi
2. P. Maillard
3. Thomas Madaule
4. L. Arguin
5. Ofer Zeitouni
6. Jian Ding
7. David Belius
8. Eliran Subag

In Mathematical Physics

1. Zhan Shi
2. Thomas Madaule
3. L. Arguin
4. Ofer Zeitouni
5. David Belius
6. Jian Ding
7. Joseph Najnudel
8. P. Maillard

Citations in the corpus

• Branching Brownian motion seen from its tip (2011, background): “In [4] a complete description of the extremal point process of the branching Brownian motion is given.”
• Convergence to a single wave in the Fisher-KPP equation (2016, background): “Building on the work of Lalley and Sellke [16], recent probabilistic analyses [1, 2, 3, 8, 7] of this particle system have identified a decorated Poisson-type point process which is the limit of the particle distribution “seen from the tip”: there is a random variable Z > 0 such that the point proce”
• Branching random walks (2015, methodology): “See also Gouéré [117] for an analysis based on [27], [28], [29] and [12], and Subag and Zeitouni [225] for a general view of decorated Poisson point processes.”
• Extreme values for two-dimensional discrete Gaussian free field (2012, background): “for the limiting law of the centered maximum. More recently, the structure of the point process of maxima of the BBM was described in great detail, in a series of papers by Arguin, Bovier and Kistler [8, 9, 7] and in a paper by A¨ıd´ekon, Berestycki, Brunet and Shi [4]. These papers describe the lim”
• Maximum of a log-correlated Gaussian field (2013, background): “As in the case of the branching Brownian motion, see [6], [7], [8] and [3], our work could also lead to the “genealogy of the extremal particles” which in our context corresponds to their spatial position.”
• The subleading order of two dimensional cover times (2014, background): “Even more challenging would be the full description of the process of covering; also in this case, the analogy with BBM suggests that regions which are missed the longest form a Poisson cluster process of random intensity [1, 3].”
• Freezing and Decorated Poisson Point Processes (2014, methodology): “2 of [6] gives exactly Condition (SUS) above.”
• Maxima of a randomized Riemann zeta function, and branching random walks (2015, background): “This type of argument can also be applied to obtain the joint distribution of the near-maxima; see, for example, [3, 4, 7].”
• A note on stable point processes occurring in branching Brownian motion (2011, background): “This question was answered in the affirmative by the author [19], and independently in the special case appearing in branching Brownian motion by Arguin, Bovier, Kistler [2, 3] and A¨ıd´ekon, Berestycki, Brunet, Shi [1].”
• Convergence in Law for the Branching Random Walk Seen from Its Tip (2011, background): “2 of Aïdékon, Beresticky, Brunet and Shi [3] (and also those of [5] and [6]).”