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Exit problems associated with affine reflection groups
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  • Published: 07 August 2008

Exit problems associated with affine reflection groups

  • Yan Doumerc1 &
  • John Moriarty2 

Probability Theory and Related Fields volume 145, pages 351–383 (2009)Cite this article

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  • 2 Citations

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Abstract

We obtain a formula for the distribution of the first exit time of Brownian motion from the alcove of an affine Weyl group. In most cases the formula is expressed compactly, in terms of Pfaffians. Expected exit times are derived in the type \({\widetilde{A}}\) case. The results extend to other Markov processes. We also give formulas for the real eigenfunctions of the Dirichlet and Neumann Laplacians on alcoves, observing that the ‘Hot Spots’ conjecture of J. Rauch is true for alcoves.

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References

  1. Alabert A., Farré M., Roy R.: Exit times from equilateral triangles. Appl. Math. Optim. 49(1), 43–53 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bañuelos R., Burdzy K.: On the “hot spots” conjecture of J. Rauch. J. Funct. Anal. 164(1), 1–33 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bass R.F.: Probabilistic techniques in analysis. probability and its applications (New York). Springer, New York (1995)

    Google Scholar 

  4. Bérard P.H.: Spectres et groupes cristallographiques. I. Domaines euclidiens. Invent. Math. 58(2), 179–199 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bruss F.T., Louchard G., Turner J.W.: On the N-tower problem and related problems. Adv. Appl. Probab. 35(1), 278–294 (2003) In honor of Joseph Mecke

    Article  MATH  MathSciNet  Google Scholar 

  6. de Bruijn N.G.: On some multiple integrals involving determinants. J. Indian Math. Soc. (N.S.) 19, 133–151 (1955)

    MATH  MathSciNet  Google Scholar 

  7. Demni, N.: Radial dunkl processes : Existence and uniqueness, hitting time, beta processes and random matrices. arXiv:0707.0367v1 [math.PR]

  8. Doumerc Y., O’Connell N.: Exit problems associated with finite reflection groups. Probab. Theory Relat. Fields. 132(4), 501–538 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Goodman R., Wallach N.R.: Representations and invariants of the classical groups, volume 68 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  10. 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)

    Article  MATH  MathSciNet  Google Scholar 

  11. Hobson D.G., Werner W.: Non-colliding Brownian motions on the circle. Bull. London Math. Soc. 28(6), 643–650 (1996)

    Article  MathSciNet  Google Scholar 

  12. Humphreys J.E.: Reflection groups and Coxeter groups, volume 29 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1990)

    Google Scholar 

  13. Kane R.: Reflection groups and invariant theory. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 5. Springer, New York (2001)

    Google Scholar 

  14. Pinsky M.A.: The eigenvalues of an equilateral triangle. SIAM J. Math. Anal. 11(5), 819–827 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  15. Stembridge J.R.: Nonintersecting paths, Pfaffians, and plane partitions. Adv. Math. 83(1), 96–131 (1990)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Classes Préparatoire, Lycée Gaston Berger, Avenue Gaston Berger, BP 69, 59016, Lille Cedex, France

    Yan Doumerc

  2. School of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, UK

    John Moriarty

Authors
  1. Yan Doumerc
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  2. John Moriarty
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Correspondence to John Moriarty.

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Cite this article

Doumerc, Y., Moriarty, J. Exit problems associated with affine reflection groups. Probab. Theory Relat. Fields 145, 351–383 (2009). https://doi.org/10.1007/s00440-008-0171-9

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  • Received: 25 June 2007

  • Revised: 02 May 2008

  • Published: 07 August 2008

  • Issue Date: November 2009

  • DOI: https://doi.org/10.1007/s00440-008-0171-9

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Keywords

  • Brownian Motion
  • Weyl Group
  • Equilateral Triangle
  • Exit Time
  • Consistent Subset
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