Probability Theory and Related Fields

, Volume 132, Issue 4, pp 501–538 | Cite as

Exit problems associated with finite reflection groups

  • Yan Doumerc
  • Neil O’Connell


We obtain a formula for the distribution of the first exit time of Brownian motion from a fundamental region associated with a finite reflection group. In the type A case it is closely related to a formula of de Bruijn and the exit probability is expressed as a Pfaffian. Our formula yields a generalisation of de Bruijn’s. We derive large and small time asymptotics, and formulas for expected first exit times. The results extend to other Markov processes. By considering discrete random walks in the type A case we recover known formulas for the number of standard Young tableaux with bounded height.


Reflection Stochastic Process Brownian Motion Probability Theory Markov Process 
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  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, 1965Google Scholar
  2. 2.
    Bañuelos, R., Smits, R.G.: Brownian motion in cones. Probab. Theory Relat. Fields 108 (3), 299–319 (1997)Google Scholar
  3. 3.
    Biane, Ph.: Minuscule weights and random walks on lattices. Quantum probability and related topics, 51–65, World Sci. Publishing, River Edge, NJ, 1992Google Scholar
  4. 4.
    Dante de Blassie, R.: Exit times from cones in ℝn of Brownian motion. Probab. Theory Relat. Fields 74 (1), 1–29 (1987)Google Scholar
  5. 5.
    Bramson, M., Griffeath, D.: Capture problems for coupled random walks. In: Random walks, Brownian motion, and interacting particle systems, 153–188, Progr. Probab., 28, Birkhäuser Boston, Boston, MA, 1991Google Scholar
  6. 6.
    de Bruijn, N.G.: On some multiple integrals involving determinants. J. Indian Math. Soc. (N.S.) 19, 133–151 (1955)Google Scholar
  7. 7.
    Burgess, D., Zhang, B.: Moments of the Lifetime of Conditioned Brownian Motion in Cones. Proc. Amer. Math. Soc. 121 (3), 925–929 (1994)Google Scholar
  8. 8.
    Burkholder, D.L.: Exit times of Brownian motion, harmonic majorization and Hardy spaces. Advances in Math. 26 (2), 182–205 (1997)Google Scholar
  9. 9.
    Comtet, A., Desbois, J.: Brownian motion in wedges, last passage time and the second arc-sine law. J. Phys. A 36, L255–L261 (2003)Google Scholar
  10. 10.
    Freidlin, M.I., Wentzell, A.D.: Random perturbations of dynamical systems. Grundlehren der Mathematischen Wissenschaften, 260. New York: Springer-Verlag, 1984Google Scholar
  11. 11.
    Gessel, I.: Symmetric functions and P-recursiveness. J. Comb. Th. Series A 53, 257–285 (1990)Google Scholar
  12. 12.
    Gessel, I., Zeilberger, D.: Random walk in a Weyl chamber. Proc. Amer. Math. Soc. 115 (1), 27–31 (1992)Google Scholar
  13. 13.
    Goodman, R., Wallach, N.: Representations and invariants of the classical groups. Encyclopedia of Mathematics and its Applications, 68. Cambridge: Cambridge University Press, 1998Google Scholar
  14. 14.
    Gordan, B.: A proof of the Bender-Knuth conjecture. Pacific J. Math. 108 (1), 99–113 (1983)Google Scholar
  15. 15.
    Grabiner, D.J.: Brownian motion in a Weyl chamber, non-colliding particles, and random matrices. Ann. Inst. H. Poincaré Probab. Statist. 35 (2), 177–204 (1999)Google Scholar
  16. 16.
    Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge University Press, 1990Google Scholar
  17. 17.
    Karlin, S., McGregor, J.: Coincidence probabilities. Pacific J. Math. 9, 1141–1164 (1959)Google Scholar
  18. 18.
    Luque, J.-G., Thibon, J.-Y.: Pfaffian and hafnian identities in shuffle algebras. Advances in Appl. Math. 29, 620-646 (2002)Google Scholar
  19. 19.
    Mehta, M.L.: Random matrices. Second edition. Academic Press, Inc., Boston, MA, 1991Google Scholar
  20. 20.
    O’Connell, N., Unwin, A.: Cones and collisions: a duality. Stoch. Proc. Appl. 43, 291 (1992)Google Scholar
  21. 21.
    Spitzer, F.: Some theorems concerning 2-dimensional Brownian motion. Trans. Amer. Math. Soc. 87, 187–197 (1958)Google Scholar
  22. 22.
    Stembridge, J.R.: Nonintersecting paths, Pfaffians, and plane partitions. Adv. Math. 83 (1), 96–131 (1990)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Yan Doumerc
    • 1
  • Neil O’Connell
    • 2
  1. 1.Institut de Mathématiques, Laboratoire de Statistique et ProbabilitésUniversité Paul-SabatierToulouseFrance
  2. 2.Mathematics InstituteUniversity of WarwickFrance

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