Advertisement

Radial Dunkl Processes Associated with Dihedral Systems

  • Nizar DemniEmail author
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 1979)

Abstract

We are interested in radial Dunkl processes associated with dihedral systems. We write down the semi-group density and as a by-product the generalized Bessel function and the W-invariant generalized Hermite polynomials. Then, a skew product decomposition, involving only independent Bessel processes, is given and the tail distribution of the first hitting time of boundary of the Weyl chamber is computed.

Keywords

Brownian Motion Coxeter Group Jacobi Polynomial Dihedral Group Weyl Chamber 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. C. Baez. The octonions. Bull. Amer. Math. Soc. (N. S.) 39, no. 2. 2002, 145–205.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    T. H. Baker, P. J. Forrester. The Calogero-Sutherland model and generalized classical polynomials. Comm. Math. Phys. 188. 1997, 175–216.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    D. Bakry. Remarques sur les semi-groupes de Jacobi. Hommage à P. A. Meyer et J. Neveu. Astérisque 236, 1996, 23–39.MathSciNetGoogle Scholar
  4. 4.
    R. Bañuelos, R. G. Smits. Brownian motions in cones. P. T. R. F. 108, 1997, 299–319.zbMATHCrossRefGoogle Scholar
  5. 5.
    O. Chybiryakov. Processus de Dunkl et relation de Lamperti. Thèse de doctorat, Université Paris VI, June 2005.Google Scholar
  6. 6.
    N. Demni, M. Zani. Large deviations for statistics of Jacobi process. To appear in S. P. A. Google Scholar
  7. 7.
    N. Demni. First hitting time of the boundary of a Weyl chamber by radial Dunkl processes. SIGMA Journal. 4, 2008, 074, 14 pages.Google Scholar
  8. 8.
    N. Demni. Generalized Bessel function of type D. SIGMA Journal. 4, 2008, 075, 7 pages.Google Scholar
  9. 9.
    N. Demni. Note on radial Dunkl processes. Submitted to Ann. I. H. P. Google Scholar
  10. 10.
    Y. Doumerc. Matrix Jacobi Process. Thèse de doctorat, Université Paul Sabatier, May 2005.Google Scholar
  11. 11.
    C. F. Dunkl, Y. Xu. Orthogonal Polynomials of Several Variables. Encyclopedia of Mathematics and Its Applications. Cambridge University Press. 2001.Google Scholar
  12. 12.
    C. F. Dunkl. Differential-difference operators associated to reflection groups. Trans. Amer. Math. Soc. 311. 1989, no. 1, 167–183.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    C. F. Dunkl. Integral kernels with reflection group invariance. Canad. J. Math. 43. 1991, no. 6, 1213–1227.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    C. F. Dunkl Generating functions associated with dihedral groups. Special functions (Hong Kong 1999), World Sci. Publ. Rier Edge, NJ. 2000, 72–87.Google Scholar
  15. 15.
    D. J. Grabiner. Brownian motion in a Weyl chamber, non-colliding particles and random matrices. Ann. IHP. 35, 1999, no. 2. 177–204.MathSciNetzbMATHGoogle Scholar
  16. 16.
    J. E. Humphreys. Reflections Groups and Coxeter Groups. Cambridge University Press. 29. 2000.Google Scholar
  17. 17.
    N. N. Lebedev. Special Functions and their Applications. Dover Publications, INC. 1972.Google Scholar
  18. 18.
    D. Revuz, M. Yor. Continuous Martingales and Brownian Motion, 3rd ed., Springer, 1999.Google Scholar
  19. 19.
    W. Schoutens. Stochastic Processes and Orthogonal Polynomials. Lecture Notes in Statistics, 146. Springer, 2000.Google Scholar
  20. 20.
    M. Rösler. Dunkl operator: theory and applications, orthogonal polynomials and special functions (Leuven, 2002). Lecture Notes in Math. Vol. 1817, Springer, Berlin, 2003, 93–135.Google Scholar
  21. 21.
    J. Warren, M. Yor. The Brownian Burglar: conditioning Brownian motion by its local time process. Sém. Probab. XXXII., 1998, 328–342.MathSciNetGoogle Scholar
  22. 22.
    M. Yor. Loi de l'indice du lacet brownien et distribution de Hartman-Watson, Zeit. Wahr. verw. Geb. 53, no. 1, 1980, 71–95.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Fakultät für mathematikuniversität BielefeldBielefeldGermany

Personalised recommendations