Advertisement

Large Deviations for the Rightmost Position in a Branching Brownian Motion

  • Bernard Derrida
  • Zhan ShiEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 208)

Abstract

We study the lower deviation probability of the position of the rightmost particle in a branching Brownian motion and obtain its large deviation function.

Keywords

Branching Brownian motion Lower deviation probability 

2010 Mathematics Subject Classification

60F10 60J80 

References

  1. 1.
    Aïdékon, E., Berestycki, J., Brunet, E., Shi, Z.: Branching Brownian motion seen from its tip. Probab. Theory Relat. Fields 157, 405–451 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arguin, L.P., Bovier, A., Kistler, N.: The extremal process of branching Brownian motion. Probab. Theory Relat. Fields 157, 535–574 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berestycki, J. Topics on Branching Brownian Motion. Lecture notes available at: http://www.stats.ox.ac.uk/~berestyc/articles.html (2015)
  4. 4.
    Bovier, A.: Gaussian Processes on Trees. Cambridge University Press, New York (2016)Google Scholar
  5. 5.
    Bramson, M.D.: Maximal displacement of branching Brownian motion. Comm. Pure Appl. Math. 31, 531–581 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bramson, M.D.: Convergence of solutions of the Kolmogorov equation to travelling waves. Mem. Amer. 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. EPL (Europhys. Lett.) 87, 60010 (2009)Google Scholar
  8. 8.
    Brunet, E., Derrida, B.: A branching random walk seen from the tip. J. Stat. Phys. 143, 420–446 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    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
  10. 10.
    Chen, X.: Waiting times for particles in a branching Brownian motion to reach the rightmost position. Stoch. Proc. Appl. 123, 3153–3182 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, 2nd edn. Springer, New York (1998)CrossRefzbMATHGoogle Scholar
  12. 12.
    Derrida, B., Meerson, B., Sasorov, P.V.: Large-displacement statistics of the rightmost particle of the one-dimensional branching Brownian motion. Phys. Rev. E 93, 042139 (2016)CrossRefGoogle Scholar
  13. 13.
    Derrida, B., Shi, Z.: Large deviations for the branching Brownian motion in presence of selection or coalescence. J. Stat. Phys. 163, 1285–1311 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Derrida, B. and Shi, Z.: Slower deviations of the branching Brownian motion and of branching random walks. J. Phys. A 50, 344001 (2017)Google Scholar
  15. 15.
    Derrida, B., Spohn, H.: Polymers on disordered trees, spin glasses, and traveling waves. J. Stat. Phys. 51, 817–840 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    den Hollander, F.: Large Deviations. American Mathematical Society, Providence (2000)zbMATHGoogle Scholar
  17. 17.
    Hu, Y., Shi, Z.: Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees. Ann. Probab. 37, 742–789 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lalley, S.P., Sellke, T.: A conditional limit theorem for the frontier of a branching Brownian motion. Ann. Probab. 15, 1052–1061 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Majumdar, S.N., Krapivsky, P.L.: Extremal paths on a random Cayley tree. Phys. Rev. E 62, 7735 (2000)CrossRefGoogle Scholar
  20. 20.
    McKean, H.P.: Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov. Commun. Pure Appl. Math. 28, 323–331 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Meerson, B., Sasorov, P.V.: Negative velocity fluctuations of pulled reaction fronts. Phys. Rev. E 84, 030101(R) (2011)CrossRefGoogle Scholar
  22. 22.
    Mueller, A.H., Munier, S.: Phenomenological picture of fluctuations in branching random walks. Phys. Rev. E 90, 042143 (2014)CrossRefGoogle Scholar
  23. 23.
    Ramola, K., Majumdar, S.N., Schehr, G.: Spatial extent of branching Brownian motion. Phys. Rev. E 91, 042131 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Rouault, A.: Large deviations and branching processes. Proceedings of the 9th International Summer School on Probability Theory and Mathematical Statistics (Sozopol, 1997). Pliska Studia Math. Bulgarica 13, 15–38 (2000)Google Scholar
  25. 25.
    Schmidt, M.A., Kistler, N.: From Derrida’s random energy model to branching random walks: from 1 to 3. Electronic Commun. Prob. 20, 1–12 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Shi, Z.: Branching Random Walks. École d’été Saint-Flour XLII (2012). Lecture Notes in Mathematics 2151. Springer, Berlin (2015)Google Scholar
  27. 27.
    Zeitouni, O.: Branching Random Walks and Gaussian Fields. Lecture notes available at: http://www.wisdom.weizmann.ac.il/~zeitouni/pdf/notesBRW.pdf (2012)

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Collège de FranceParis Cedex 05France
  2. 2.Université Pierre et Marie CurieParis Cedex 05France

Personalised recommendations