Electronic Journal of Probability

The Maximum of Brownian Motion with Parabolic Drift

Svante Janson, Guy Louchard, and Anders Martin-Löf

Full-text: Open access


We study the maximum of a Brownian motion with a parabolic drift; this is a random variable that often occurs as a limit of the maximum of discrete processes whose expectations have a maximum at an interior point. We give new series expansions and integral formulas for the distribution and the first two moments, together with numerical values to high precision.

Article information

Electron. J. Probab., Volume 15 (2010), paper no. 61, 1893-1929.

Accepted: 17 November 2010
First available in Project Euclid: 1 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J65: Brownian motion [See also 58J65]

Brownian motion parabolic drift Airy functions

This work is licensed under aCreative Commons Attribution 3.0 License.


Janson, Svante; Louchard, Guy; Martin-Löf, Anders. The Maximum of Brownian Motion with Parabolic Drift. Electron. J. Probab. 15 (2010), paper no. 61, 1893--1929. doi:10.1214/EJP.v15-830. https://projecteuclid.org/euclid.ejp/1464819846

Export citation


  • M. Abramowitz, I. A. ; Stegun, eds., Handbook of Mathematical Functions}. Dover, New York, 1972.
  • Albright, J. R. Integrals of products of Airy functions. J. Phys. A 10 (1977), no. 4, 485–490.
  • J. R. Albright ; E. P. Gavathas, Integrals involving Airy functions. Comment on: “Evaluation of an integral involving Airy functions” by L. T. Wille and J. Vennik. J. Phys. A vol 19d (1986), no. 13, 2663–2665.
  • Barbour, Andrew D. A note on the maximum size of a closed epidemic. J. Roy. Statist. Soc. Ser. B 37 (1975), no. 3, 459–460.
  • Barbour, A. D. Brownian motion and a sharply curved boundary. Adv. in Appl. Probab. 13 (1981), no. 4, 736–750.
  • Daniels, H. E. The maximum size of a closed epidemic. Advances in Appl. Probability 6 (1974), 607–621.
  • Daniels, H. E. The maximum of a Gaussian process whose mean path has a maximum, with an application to the strength of bundles of fibres. Adv. in Appl. Probab. 21 (1989), no. 2, 315–333.
  • Daniels, H. E.; Skyrme, T. H. R. The maximum of a random walk whose mean path has a maximum. Adv. in Appl. Probab. 17 (1985), no. 1, 85–99.
  • Groeneboom, P. Estimating a monotone density. Proceedings of the Berkeley conference in honor of Jerzy Neyman and Jack Kiefer, Vol. II (Berkeley, Calif., 1983), 539–555, Wadsworth Statist./Probab. Ser., Wadsworth, Belmont, CA, 1985.
  • Groeneboom, Piet. Brownian motion with a parabolic drift and Airy functions. Probab. Theory Related Fields 81 (1989), no. 1, 79–109.
  • Groeneboom, Piet. Brownian motion with a parabolic drift and Airy functions. Probab. Theory Related Fields 81 (1989), no. 1, 79–109.
  • P. Groeneboom ; J. A. Wellner. Computing Chernoff's distribution. J. Comput. Graph. Statist. vol 10 (2001), no. 2, 388–400.
  • Hörmander, Lars. The analysis of linear partial differential operators. I.Distribution theory and Fourier analysis.Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 256. Springer-Verlag, Berlin, 1983. ix+391 pp. ISBN: 3-540-12104-8 (85g:35002a)
  • Janson, Svante. Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas. Probab. Surv. 4 (2007), 80–145 (electronic).
  • Janson, Svante. Sorting using complete subintervals and the maximum number of runs in a randomly evolving sequence. Ann. Comb. 12 (2009), no. 4, 417–447. (Review)
  • Lachal, Aimé. Sur la distribution de certaines fonctionnelles de l'intégrale du mouvement brownien avec dérives parabolique et cubique.(French) [Distribution of certain functionals of the integral of Brownian motion with parabolic and cubic drift] Comm. Pure Appl. Math. 49 (1996), no. 12, 1299–1338.
  • Lefebvre, Mario. First-passage densities of a two-dimensional process. SIAM J. Appl. Math. 49 (1989), no. 5, 1514–1523.
  • G. Louchard, Random walks, Gaussian processes and list structures. Theoret. Comput. Sci. 53 (1987), no. 1, 99–124.
  • Louchard, G.; Kenyon, Claire; Schott, R. Data structures' maxima. SIAM J. Comput. 26 (1997), no. 4, 1006–1042.
  • Martin-Löf, Anders. The final size of a nearly critical epidemic, and the first passage time of a Wiener process to a parabolic barrier. J. Appl. Probab. 35 (1998), no. 3, 671–682.
  • Salminen, Paavo. On the first hitting time and the last exit time for a Brownian motion to/from a moving boundary. Adv. in Appl. Probab. 20 (1988), no. 2, 411–426.
  • Smith, Richard L. The asymptotic distribution of the strength of a series-parallel system with equal load-sharing. Ann. Probab. 10 (1982), no. 1, 137–171.
  • Steinsaltz, David. Random time changes for sock-sorting and other stochastic process limit theorems. Electron. J. Probab. 4 (1999), no. 14, 25 pp. (electronic).