Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Slowdown in branching Brownian motion with inhomogeneous variance

Pascal Maillard and Ofer Zeitouni

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We consider the distribution of the maximum $M_{T}$ of branching Brownian motion with time-inhomogeneous variance of the form $\sigma^{2}(t/T)$, where $\sigma(\cdot)$ is a strictly decreasing function. This corresponds to the study of the time-inhomogeneous Fisher–Kolmogorov–Petrovskii–Piskunov (F-KPP) equation $F_{t}(x,t)=\sigma^{2}(1-t/T)F_{xx}(x,t)/2+g(F(x,t))$, for appropriate nonlinearities $g(\cdot)$. Fang and Zeitouni (J. Stat. Phys. 149 (2012) 1–9) showed that $M_{T}-v_{\sigma}T$ is negative of order $T^{1/3}$, where $v_{\sigma}=\int_{0}^{1}\sigma(s)\,\mathrm{d}s$. In this paper, we show the existence of a function $m'_{T}$, such that $M_{T}-m'_{T}$ converges in law, as $T\rightarrow\infty$. Furthermore, $m'_{T}=v_{\sigma}T-w_{\sigma}T^{1/3}-\sigma(1)\log T+O(1)$ with $w_{\sigma}=2^{-1/3}\alpha_{1}\int_{0}^{1}\sigma(s)^{1/3}|\sigma'(s)|^{2/3}\,\mathrm{d}s$. Here, $-\alpha_{1}=-2.33811\ldots$ is the largest zero of the Airy function $\operatorname{Ai}$. The proof uses a mixture of probabilistic and analytic arguments.


Nous étudions la loi du maximum $M_{T}$ d’un mouvement brownien branchant avec une variance inhomogène en temps de la form $\sigma^{2}(t/T)$, où $\sigma(\cdot)$ est une fonction strictement décroissante. Ceci correspond à étudier l’équation Fisher–Kolmogorov–Petrovskii–Piskunov (F-KPP) inhomogène en temps, $F_{t}(x,t)=\sigma^{2}(1-t/T)F_{xx}(x,t)/2+g(F(x,t))$, pour des nonlinéarités $g(\cdot)$ appropriées. Fang et Zeitouni (J. Stat. Phys. 149 (2012) 1–9) ont montré que $M_{T}-v_{\sigma}T$ est negatif de l’ordre $T^{1/3}$, où $v_{\sigma}=\int_{0}^{1}\sigma(s)\,\mathrm{d}s$. Dans cet article, nous montrons l’existence d’une fonction $m'_{T}$ telle que $M_{T}-m'_{T}$ converge en loi quand $T\rightarrow\infty$. De plus, $m'_{T}=v_{\sigma}T-w_{\sigma}T^{1/3}-\sigma(1)\log T+O(1)$ avec $w_{\sigma}=2^{-1/3}\alpha_{1}\int_{0}^{1}\sigma(s)^{1/3}|\sigma'(s)|^{2/3}\,\mathrm{d}s$. Ici, $-\alpha_{1}=-2.33811\ldots$ est la plus grande racine de la fonction d’Airy $\operatorname{Ai}$. La démonstration repose sur un mélange d’arguments probabilistes et analytiques.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 3 (2016), 1144-1160.

Received: 3 February 2014
Revised: 6 March 2015
Accepted: 6 March 2015
First available in Project Euclid: 28 July 2016

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Maillard, Pascal; Zeitouni, Ofer. Slowdown in branching Brownian motion with inhomogeneous variance. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 3, 1144--1160. doi:10.1214/15-AIHP675.

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  • [1] E. Aïdékon. Convergence in law of the minimum of a branching random walk. Ann. Probab. 41 (3A) (2013) 1362–1426.
  • [2] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series 55. U.S. Government Printing Office, Washington, DC, 1964.
  • [3] A. Bovier and I. Kurkova. Derrida’s generalized random energy models. II. Models with continuous hierarchies. Ann. Inst. Henri Poincaré Probab. Stat. 40 (2004) 481–495.
  • [4] M. Bramson. Maximal displacement of branching Brownian motion. Comm. Pure Appl. Math. 31 (1978) 531–581.
  • [5] M. Bramson. Convergence of solutions of the Kolmogorov equation to travelling waves. Mem. Amer. Math. Soc. 44 (1983) 1–190.
  • [6] M. Bramson, J. Ding and O. Zeitouni. Convergence in law of the maximum of the two-dimensional discrete Gaussian free field, 2013. Available at arXiv:1301.6669.
  • [7] J. L. Doob. Classical Potential Theory and Its Probabilistic Counterpart. Grundlehren der Mathematischen Wissenschaften 262. Springer, New York, 1984.
  • [8] B. Derrida and H. Spohn. Polymers on disordered trees, spin glasses, and traveling waves. J. Stat. Phys. 51 (1988) 817–840.
  • [9] L. C. Evans. Partial Differential Equations. Graduate Studies in Mathematics 19. Amer. Math. Soc., Providence, RI, 1998.
  • [10] M. Fang and O. Zeitouni. Slowdown for time inhomogeneous branching Brownian motion. J. Stat. Phys. 149 (2012) 1–9.
  • [11] P. Ferrari and H. Spohn. Constrained Brownian motion: Fluctuations away from circular and parabolic barriers. Ann. Probab. 33 (2005) 1302–1325.
  • [12] C. W. Gardiner. Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 2nd edition. Springer Series in Synergetics 13. Springer, Berlin, 1985.
  • [13] P. Groeneboom. Brownian motion with a parabolic drift and Airy functions. Probab. Theory Related Fields 81 (1989) 79–109.
  • [14] N. Ikeda, M. Nagasawa and S. Watanabe. Branching Markov processes III. J. Math. Kyoto Univ. 9 (1) (1969) 95–160.
  • [15] S. Lalley and T. Sellke. A conditional limit theorem for the frontier of a branching Brownian motion. Ann. Probab. 15 (3) (1987) 1052–1061.
  • [16] T. Madaule. First order transition for the branching random walk at the critical parameter, 2012. Available at arXiv:1206.3835.
  • [17] B. Mallein. Maximal displacement of a branching random walk in time-inhomogeneous environment, 2013. Available at arXiv:1307.4496.
  • [18] H. McKean. Application of Brownian motion to the equation of Kolmogorov–Petrovskii–Piskunov. Comm. Pure Appl. Math. 28 (1975) 323–331.
  • [19] J. Neveu. Multiplicative martingales for spatial branching processes. In Seminar on Stochastic Processes (Princeton, NJ, 1987). Progress in Probability and Statistics 15 223–242. Birkhäuser, Boston, MA, 1988.
  • [20] J. Nolen, J.-M. Roquejoffre and L. Ryzhik. Power-like delay in time inhomogeneous Fisher–KPP equations. Comm. Partial Differential Equations 40 (2015) 475–505.
  • [21] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Springer, Berlin, 1999.
  • [22] M. I. Roberts. A simple path to asymptotics for the frontier of a branching Brownian motion. Ann. Probab. 41 (2013) 3518–3541.
  • [23] D. Slepian. The one-sided barrier problem for Gaussian noise. Bell System Tech. J. 41 (1962) 463–501.
  • [24] O. Vallée and M. Soares. Airy Functions and Applications to Physics. Imperial College Press, London, 2004.
  • [25] T. Yang and Y.-X. Ren. Limit theorem for derivative martingale at criticality w.r.t. branching Brownian motion. Statist. Probab. Lett. 81 (2) (2011) 195–200.