Advertisement

Probability Theory and Related Fields

, Volume 157, Issue 1–2, pp 405–451 | Cite as

Branching Brownian motion seen from its tip

  • E. Aïdékon
  • J. Berestycki
  • É. Brunet
  • Z. Shi
Article

Abstract

It has been conjectured since the work of Lalley and Sellke (Ann. Probab., 15, 1052–1061, 1987) that branching Brownian motion seen from its tip (e.g. from its rightmost particle) converges to an invariant point process. Very recently, it emerged that this can be proved in several different ways (see e.g. Brunet and Derrida, A branching random walk seen from the tip, 2010, Poissonian statistics in the extremal process of branching Brownian motion, 2010; Arguin et al., The extremal process of branching Brownian motion, 2011). The structure of this extremal point process turns out to be a Poisson point process with exponential intensity in which each atom has been decorated by an independent copy of an auxiliary point process. The main goal of the present work is to give a complete description of the limit object via an explicit construction of this decoration point process. Another proof and description has been obtained independently by Arguin et al. (The extremal process of branching Brownian motion, 2011).

Mathematics Subject Classification

60J80 60G70 

Notes

Acknowledgments

We wish to warmly thank two anonymous referees for their careful reading and fruitful suggestions. We also wish to express our gratitude to Louis-Pierre Arguin for a useful conversation and Henri Berestycki for the argument behind Remark 6.3.

References

  1. 1.
    Aïdékon, E.: Convergence in law of the minimum of a branching random walk. Ann. Probab. ArXiv 1101.1810 [math.PR] (to appear)Google Scholar
  2. 2.
    Arguin, L.-P., Bovier, A., Kistler, N.: The genealogy of extremal particles of branching Brownian motion. ArXiv 1008.4386 [math.PR] (2010)Google Scholar
  3. 3.
    Arguin, L.-P., Bovier, A., Kistler, N.: Poissonian statistics in the extremal process of branching Brownian motion. ArXiv 1010.2376 [math.PR] (2010)Google Scholar
  4. 4.
    Arguin, L.-P., Bovier, A., Kistler, N.: The extremal process of branching Brownian motion. ArXiv 1103.2322 [math.PR] (2011)Google Scholar
  5. 5.
    Bramson, M.: Maximal displacement of branching Brownian motion. Commun. Pure Appl. Math. 31(5), 531–581 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bramson, M.: Convergence of solutions of the Kolmogorov equation to travelling waves. Mem. Am. Math. Soc. 44(285) (1983)Google Scholar
  7. 7.
    Brunet, E., Derrida, B.: Statistics at the tip of a branching random walk and the delay of traveling waves. Eur. Phys. Lett. 87, 60010 (2009)CrossRefGoogle Scholar
  8. 8.
    Brunet, E., Derrida, B.: A branching random walk seen from the tip. ArXiv cond-mat/1011.4864 (2010)Google Scholar
  9. 9.
    Chauvin, B., Rouault, A.: Étude de l’équation KPP et du branchement brownien en zones sous-critique et critique. Application aux arbres spatiaux. C. R Acad. Sci. Paris Sér. I Math. 304, 19–22 (1987)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Chauvin, B., Rouault, A.: KPP equation and supercritical branching Brownian motion in the subcritical speed area. Application to spatial trees. Probab. Theory Relat. Fields 80, 299–314 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chauvin, B.: Multiplicative martingales and stopping lines for branching Brownian motion. Ann. Probab. 30:1195–1205 (1991)Google Scholar
  12. 12.
    Denisov, I.V.: A random walk and a Wiener process near a maximum. Theory Probab. Appl. 28, 821–824 (1983)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs (1964)zbMATHGoogle Scholar
  14. 14.
    Hardy, R., Harris, S.C.: A new formulation of the spine approach to branching diffusions. arXiv: 0611.054 [math.PR] (2006)Google Scholar
  15. 15.
    Harris, S.C.: Travelling-waves for the FKPP equation via probabilistic arguments. Proc. R. Soc. Edinb. 129A, 503–517 (1999)CrossRefGoogle Scholar
  16. 16.
    Harris, J.W., Harris, S.C., Kyprianou, A.E.: Further probabilistic analysis of the Fisher–Kolmogorov–Petrovskii–Piscounov equation: one-sided traveling waves. Ann. Inst. H. Poincaré Probab. Stat. 42, 125–145 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Harris, S.C., Roberts, M.I.: The many-to-few lemma and multiple spines. arXiv:1106.4761 [math.PR] (2011)Google Scholar
  18. 18.
    Ikeda, N., Nagasawa, M., Watanabe, S.: On branching Markov processes. Proc. Japan Acad. 41, 816–821 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Imhof, J.-P.: Density factorizations for Brownian motion, meander and the three-dimensional Bessel process, and applications. J. Appl. Probab. 21, 500–510 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Jagers, P.: General branching processes as Markov fields. Stoch. Process. Appl. 32, 193–212 (1989)MathSciNetGoogle Scholar
  21. 21.
    Kallenberg, O.: Random measures. Springer, Berlin (1983)zbMATHGoogle Scholar
  22. 22.
    Kyprianou, A.: Travelling wave solutions to the K-P-P equation: alternatives to Simon Harris’ probabilistic analysis. Ann. Inst. H. Poincaré Probab. Statist. 40, 53–72 (2004)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Lalley, S.P., Sellke, T.: A conditional limit theorem for frontier of a branching Brownian motion. Ann. Probab. 15, 1052–1061 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    McKean, H.P.: Application of Brownian motion to the equation of Kolmogorov–Petrovskii–Piskunov. Commun. Pure Appl. Math. 28, 323–331 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Madaule, T.: Convergence in law for the branching random walk seen from its tip. arXiv:1107.2543 [math.PR] (2011)Google Scholar
  26. 26.
    Maillard, P.: A characterisation of superposable random measures. arXiv:1102.1888 [math.PR] (2011)Google Scholar
  27. 27.
    Maillard, P.: Branching Brownian motion with selection of the \(N\) right-mostparticles: an approximate model. arXiv:1112.0266v2 [math.PR] (2012)Google Scholar
  28. 28.
    Neveu, J.: Multiplicative martingales for spatial branching processes. Seminar on Stochastic Processes, Princeton, pp 223–242. Progr. Probab. Statist., vol. 15, Birkhäuser, Boston (1988)Google Scholar
  29. 29.
    Resnick, S.I.: Extreme values, regular variation, and point processes. Applied Probability Series, vol. 4, Springer, Berlin (1987)Google Scholar
  30. 30.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin (1999)CrossRefzbMATHGoogle Scholar
  31. 31.
    Williams, D.: Path decomposition and continuity of local time for one-dimensional diffusions. I. Proc. Lond. Math. Soc. (3) 28, 738–768 (1974)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • E. Aïdékon
    • 1
  • J. Berestycki
    • 2
  • É. Brunet
    • 3
  • Z. Shi
    • 2
  1. 1.Department of Mathematics and Computer ScienceTechnische Universiteit EindhovenEindhovenThe Netherlands
  2. 2.Laboratoire de Probabilités et Modèles AléatoiresCNRS UMR 7599, UPMC Université Paris 6 Paris Cedex 05France
  3. 3.Laboratoire de Physique Statistique, École Normale SupérieureUPMC Université Paris 6, Université Paris Diderot, CNRS ParisFrance

Personalised recommendations