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Large Deviations for the Branching Brownian Motion in Presence of Selection or Coalescence

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Abstract

The large deviation function has been known for a long time in the literature for the displacement of the rightmost particle in a branching random walk (BRW), or in a branching Brownian motion (BBM). More recently a number of generalizations of the BBM and of the BRW have been considered where selection or coalescence mechanisms tend to limit the exponential growth of the number of particles. Here we try to estimate the large deviation function of the position of the rightmost particle for several such generalizations: the L-BBM, the N-BBM, and the coalescing branching random walk (CBRW) which is closely related to the noisy FKPP equation. Our approach allows us to obtain only upper bounds on these large deviation functions. One noticeable feature of our results is their non analytic dependence on the parameters (such as the coalescence rate in the CBRW).

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Notes

  1. Although the right-hand side of (4.1) is an event of the BBM, not of the L-BBM, we use the superscript in \(\widetilde{E}_t^{\mathrm {LBBM}}\) to remind us that it will serve to study the large deviation function for the L-BBM. A similar remark applies to the forthcoming events \(\widetilde{E}_t^{\mathrm {NBBM}}\) and \(\widetilde{E}_t^{\mathrm {CBRW}}\).

  2. The choice of the power 2 / 3 is arbitrary; anything in \((\frac{1}{2}, \, 1)\) will do the job.

  3. The choice of 3 on the right-hand side is arbitrary; anything in \((2, \, \infty )\) will do the job.

  4. The choice of powers in \((\ln N)^2\) and \((\ln N)^3\) are arbitrary: they can be replaced by \(C_1 \ln N\) and \(C_2 \ln N\) with two sufficiently large constants \(C_1\) and \(C_2\).

  5. As we shall point out in Sect. 5, this picture is probably inaccurate, and is only due to the fact that our upper bound for \(Q(x, \, s)\) is not optimal. We conjecture that regardless of the value of v, the subtrees never make any particular effort in the N-BBM, which would be in complete contrast with the L-BBM.

References

  1. Aïdékon, E.: Convergence in law of the minimum of a branching random walk. Ann. Probab. 41, 1362–1426 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bérard, J., Gouéré, J.B.: Brunet–Derrida behavior of branching-selection particle systems on the line. Commun. Math. Phys. 298, 323–342 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bérard, J., Maillard, P.: The limiting process of \(N\)-particle branching random walk with polynomial tails. Electron. J. Probab. 19, 1–17 (2014)

    MathSciNet  MATH  Google Scholar 

  4. Berestycki, J. (2015). Topics on branching Brownian motion. Lecture notes. http://www.stats.ox.ac.uk/berestyc/articles.html

  5. Berestycki, N., Zhao, L. Z., (2013). The shape of multidimensional Brunet–Derrida particle systems. arXiv preprint arXiv:1305.0254

  6. Biggins, J.D.: Large deviations and branching processes. Chernoff’s theorem in the branching random walk. J. Appl. Probab. 14, 630–636 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bramson, M.D.: Maximal displacement of branching Brownian motion. Commun. Pure Appl. Math. 31, 531–581 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bramson, M.D.: Convergence of Solutions of the Kolmogorov Equation to Travelling Waves, vol. 44, vol. 285. American Mathematical Society, Providence (1983)

    MATH  Google Scholar 

  9. Brunet, E., Derrida, B.: Shift in the velocity of a front due to a cutoff. Phys. Rev. E 56, 2597 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  10. Brunet, E., Derrida, B.: Effect of microscopic noise on front propagation. J. Stat. Phys. 103, 269–282 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Brunet, E., Derrida, B., Mueller, A.H., Munier, S.: Noisy traveling waves: effect of selection on genealogies. Europhys. Lett. 76, 1 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  12. Brunet, E., Derrida, B., Mueller, A.H., Munier, S.: Phenomenological theory giving the full statistics of the position of fluctuating pulled fronts. Phys. Rev. E 73, 056126 (2006)

    Article  ADS  Google Scholar 

  13. Brunet, E., Derrida, B., Mueller, A.H., Munier, S.: Effect of selection on ancestry: an exactly soluble case and its phenomenological generalization. Phys. Rev. E 76, 041104 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  14. 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)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chen, X. (2013). Scaling limit of the path leading to the leftmost particle in a branching random walk. arXiv preprint arXiv:1305.6723

  16. Conlon, J.G., Doering, C.R.: On travelling waves for the stochastic Fisher–Kolmogorov–Petrovsky–Piscunov equation. J. Stat. Phys. 120, 421–477 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications. Springer Science & Business Media, Berlin (2009)

    MATH  Google Scholar 

  18. Den Hollander, F.: Large Deviations. American Mathematical Society, Providence (2008)

    MATH  Google Scholar 

  19. Doering, C.R., Mueller, C., Smereka, P.: Interacting particles, the stochastic Fisher–Kolmogorov–Petrovsky–Piscounov equation, and duality. Phys. A 325, 243–259 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Durrett, R., Remenik, D.: Brunet–Derrida particle systems, free boundary problems and Wiener–Hopf equations. Ann. Probab. 39, 2043–2078 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. McKean, H.P.: Application of brownian motion to the equation of Kolmogorov–Petrovskii–Piskunov. Commun. Pure Appl. Math. 28, 323–331 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  22. Maillard, P. (2013). Speed and fluctuations of \(N\)-particle branching Brownian motion with spatial selection. arXiv preprint arXiv:1304.0562

  23. Mallein, B. (2015). Branching random walk with selection at critical rate. arXiv preprint arXiv:1502.07390

  24. Meerson, B., Sasorov, P.V.: Negative velocity fluctuations of pulled reaction fronts. Phys. Rev. E 84, 030101 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  25. Meerson, B., Vilenkin, A., Sasorov, P.V.: Emergence of fluctuating traveling front solutions in macroscopic theory of noisy invasion fronts. Phys. Rev. E 87, 012117 (2013)

    Article  ADS  Google Scholar 

  26. Mueller, C., Mytnik, L., Quastel, J.: Effect of noise on front propagation in reaction-diffusion equations of KPP type. Invent. Math. 184, 405–453 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Mueller, C., Sowers, R.B.: Random travelling waves for the KPP equation with noise. J. Funct. Anal. 128, 439–498 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  28. Pain, M. (2015). Velocity of the \(L\)-branching Brownian motion. arXiv preprint arXiv:1510.02683

  29. Panja, D.: Effects of fluctuations on propagating fronts. Phys. Rep. 393, 87–174 (2004)

    Article  ADS  Google Scholar 

  30. Pechenik, L., Levine, H.: Interfacial velocity corrections due to multiplicative noise. Phys. Rev. E 59, 3893 (1999)

    Article  ADS  Google Scholar 

  31. Ramola, K., Majumdar, S.N., Schehr, G.: Spatial extent of branching Brownian motion. Phys. Rev. E 91, 042131 (2015)

    Article  ADS  Google Scholar 

  32. Rouault, A. (2000). Large deviations and branching processes. In Proceedings of the 9th International Summer School on Probability Theory and Mathematical Statistics (Sozopol, 1997). Pliska Studia Mathematica Bulgarica 13, pp. 15–38

  33. Shi, Z., (2015). Branching random walks. École d’été Saint-Flour XLII. Lecture Notes in Mathematics 2151. Springer, Berlin (2012)

  34. van Saarloos, W.: Front propagation into unstable states. Phys. Rep. 386(2), 29–222 (2003)

    Article  ADS  MATH  Google Scholar 

  35. Zeitouni, O. (2012). Branching random walks and Gaussian fields. Lecture notes. http://www.wisdom.weizmann.ac.il/zeitouni/pdf/notesBRW

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Acknowledgments

B.D. thanks the LPMA in Jussieu for its hospitality during the whole academic year 2014–2015.

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Authors and Affiliations

  1. Collège de France, 11 place Marcelin Berthelot, 75231, Paris Cedex 05, France

    Bernard Derrida

  2. Laboratoire de Physique Statistique, École Normale Supérieure, CNRS, Université Pierre et Marie Curie, Université Denis Diderot, 24 rue Lhomond, 75231, Paris Cedex 05, France

    Bernard Derrida

  3. LPMA, Université Pierre et Marie Curie, 4 place Jussieu, 75252, Paris Cedex 05, France

    Zhan Shi

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  1. Bernard Derrida
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Derrida, B., Shi, Z. Large Deviations for the Branching Brownian Motion in Presence of Selection or Coalescence. J Stat Phys 163, 1285–1311 (2016). https://doi.org/10.1007/s10955-016-1522-z

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