## Electronic Journal of Probability

### Fine asymptotics for the consistent maximal displacement of branching Brownian motion

Matthew Roberts

#### Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465067134

Digital Object Identifier
doi:10.1214/EJP.v20-2912

Mathematical Reviews number (MathSciNet)
MR3325098

Zentralblatt MATH identifier
1320.60145

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

#### Citation

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

#### References

• Berestycki, Julien; Berestycki, NathanaÃ«l; 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, phÃ‰tude de l'équation de la diffusion avec croissance de la quantité de matiÃ¨re 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.