Electronic Journal of Probability

The extremal process of two-speed branching Brownian motion

Anton Bovier and Lisa Hartung

Full-text: Open access

Abstract

We construct and describe the extremal process for variable speed branching Brownian motion, studied recently by Fang and Zeitouni, for the case of piecewise constant speeds; in fact for simplicity we concentrate on the case when the speed is $\sigma_1$ for $s\leq bt$ and $\sigma_2$ when $bt\leq s\leq t$. In the case $\sigma_1>\sigma_2$, the process is the concatenation of two BBM extremal processes, as expected. In the case $\sigma_1<\sigma_2$, a new family  of cluster point processes arises, that are similar, but distinctively different from the BBM process. Our proofs follow the strategy of Arguin, Bovier, and Kistler.

Article information

Source
Electron. J. Probab., Volume 19 (2014), paper no. 18, 28 pp.

Dates
Accepted: 3 February 2014
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465065660

Digital Object Identifier
doi:10.1214/EJP.v19-2982

Mathematical Reviews number (MathSciNet)
MR3164771

Zentralblatt MATH identifier
1288.60108

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

Keywords
branching Brownian motion extremal processes extreme values F-KPP equation cluster point processes

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Bovier, Anton; Hartung, Lisa. The extremal process of two-speed branching Brownian motion. Electron. J. Probab. 19 (2014), paper no. 18, 28 pp. doi:10.1214/EJP.v19-2982. https://projecteuclid.org/euclid.ejp/1465065660


Export citation

References

  • Aïdékon, E.; Berestycki, J.; Brunet, É; Shi, Z. Branching Brownian motion seen from its tip. Probab. Theory Related Fields 157 (2013), no. 1-2, 405–451.
  • Arguin, Louis-Pierre; Bovier, Anton; Kistler, Nicola. Poissonian statistics in the extremal process of branching Brownian motion. Ann. Appl. Probab. 22 (2012), no. 4, 1693–1711.
  • Arguin, Louis-Pierre; Bovier, Anton; Kistler, Nicola. The extremal process of branching Brownian motion. Probab. Theory Related Fields 157 (2013), no. 3-4, 535–574.
  • Bovier, Anton; Kurkova, Irina. Derrida's generalised random energy models. I. Models with finitely many hierarchies. Ann. Inst. H. Poincaré Probab. Statist. 40 (2004), no. 4, 439–480.
  • Bovier, Anton; Kurkova, Irina. Derrida's generalized random energy models. II. Models with continuous hierarchies. Ann. Inst. H. Poincaré Probab. Statist. 40 (2004), no. 4, 481–495.
  • Bramson, Maury. Convergence of solutions of the Kolmogorov equation to travelling waves. Mem. Amer. Math. Soc. 44 (1983), no. 285, iv+190 pp.
  • Bramson, Maury D. Maximal displacement of branching Brownian motion. Comm. Pure Appl. Math. 31 (1978), no. 5, 531–581.
  • Chauvin, B.; Rouault, A. Supercritical branching Brownian motion and K-P-P equation in the critical speed-area. Math. Nachr. 149 (1990), 41–59.
  • Derrida, B.; Spohn, H. Polymers on disordered trees, spin glasses, and traveling waves. New directions in statistical mechanics (Santa Barbara, CA, 1987). J. Statist. Phys. 51 (1988), no. 5-6, 817–840.
  • Fang, Ming; Zeitouni, Ofer. Branching random walks in time inhomogeneous environments. Electron. J. Probab. 17 (2012), no. 67, 18 pp.
  • Fang, Ming; Zeitouni, Ofer. Slowdown for time inhomogeneous branching Brownian motion. J. Stat. Phys. 149 (2012), no. 1, 1–9.
  • E. Gardner and B. Derrida, phSolution of the generalised random energy model, J. Phys. C 19 (1986), 2253–2274.
  • J.-B. Gouéré, Branching Brownian motion seen from its left-most particule, ArXiv e-prints (2013).
  • Lisa Hartung, in preparation, Tech. report, 2014.
  • Lalley, S. P.; Sellke, T. A conditional limit theorem for the frontier of a branching Brownian motion. Ann. Probab. 15 (1987), no. 3, 1052–1061.
  • P. Maillard and O. Zeitouni, Slowdown in branching Brownian motion with inhomogeneous variance, ArXiv e-prints (2013).
  • B. Mallein, Maximal displacement of a branching random walk in time-inhomogeneous environment, ArXiv e-prints (2013).
  • McKean, H. P. Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov. Comm. Pure Appl. Math. 28 (1975), no. 3, 323–331.