Electronic Journal of Probability

Fine asymptotics for the consistent maximal displacement of branching Brownian motion

Matthew Roberts

Full-text: Open access


It is well-known that the maximal particle in a branching Brownian motion sits near $\sqrt2 t - \frac{3}{2\sqrt2}\log t$ at time $t$. One may then ask about the paths of particles near the frontier: how close can they stay to this critical curve? Two different approaches to this question have been developed. We improve upon the best-known bounds in each case, revealing new qualitative features including marked differences between the two approaches.

Article information

Electron. J. Probab., Volume 20 (2015), paper no. 28, 26 pp.

Accepted: 13 March 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Branching Brownian motion consistent minimal displacement survival probability growth rate

This work is licensed under aCreative Commons Attribution 3.0 License.


Roberts, Matthew. Fine asymptotics for the consistent maximal displacement of branching Brownian motion. Electron. J. Probab. 20 (2015), paper no. 28, 26 pp. doi:10.1214/EJP.v20-2912. https://projecteuclid.org/euclid.ejp/1465067134

Export citation


  • Berestycki, Julien; Berestycki, Nathanaël; Schweinsberg, Jason. Critical branching Brownian motion with absorption: survival probability. Probab. Theory Related Fields 160 (2014), no. 3-4, 489–520.
  • Bolthausen, Erwin; Deuschel, Jean-Dominique; Giacomin, Giambattista. Entropic repulsion and the maximum of the two-dimensional harmonic crystal. Ann. Probab. 29 (2001), no. 4, 1670–1692.
  • 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; Zeitouni, Ofer. Tightness of the recentered maximum of the two-dimensional discrete Gaussian free field. Comm. Pure Appl. Math. 65 (2012), no. 1, 1–20.
  • Derrida, B.; Simon, D. The survival probability of a branching random walk in presence of an absorbing wall. Europhys. Lett. EPL 78 (2007), no. 6, Art. 60006, 6 pp.
  • Fang, Ming; Zeitouni, Ofer. Consistent minimal displacement of branching random walks. Electron. Commun. Probab. 15 (2010), 106–118.
  • Faraud, Gabriel; Hu, Yueyun; Shi, Zhan. Almost sure convergence for stochastically biased random walks on trees. Probab. Theory Related Fields 154 (2012), no. 3-4, 621–660.
  • S. C. Harris and M. I. Roberts, The many-to-few lemma and multiple spines, Submitted. Preprint: http://arxiv.org/abs/1106.4761.
  • Harris, Simon C.; Roberts, Matthew I. The unscaled paths of branching Brownian motion. Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012), no. 2, 579–608.
  • Hu, Yueyun; Shi, Zhan. Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees. Ann. Probab. 37 (2009), no. 2, 742–789.
  • Jaffuel, Bruno. The critical barrier for the survival of branching random walk with absorption. Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012), no. 4, 989–1009.
  • A. N. Kolmogorov, I. Petrovski, and N. Piscounov, phÉtude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problem biologique, Mosc. Univ. Bull. Math. 1 (1937), 1–25, Translated and reprinted in Pelce, P., Dynamics of Curved Fronts (Academic, San Diego, 1988).
  • Lawler, Gregory F. Introduction to stochastic processes. Second edition. Chapman & Hall/CRC, Boca Raton, FL, 2006. xiv+234 pp. ISBN: 978-1-58488-651-8; 1-58488-651-X.
  • Neveu, J. Multiplicative martingales for spatial branching processes. Seminar on Stochastic Processes, 1987 (Princeton, NJ, 1987), 223–242, Progr. Probab. Statist., 15, Birkhäuser Boston, Boston, MA, 1988.
  • A.A. Novikov, On estimates and the asymptotic behavior of nonexit probabilities of a Wiener process to a moving boundary, Math. USSR Sbornik 38 (1981), no. 4, 495–505.
  • Roberts, Matthew I. A simple path to asymptotics for the frontier of a branching Brownian motion. Ann. Probab. 41 (2013), no. 5, 3518–3541.