Séminaire de Probabilités XLII pp 153-169 | Cite as
Radial Dunkl Processes Associated with Dihedral Systems
Chapter
First Online:
- 2 Citations
- 831 Downloads
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.
References
- 1.J. C. Baez. The octonions. Bull. Amer. Math. Soc. (N. S.) 39, no. 2. 2002, 145–205.MathSciNetzbMATHCrossRefGoogle Scholar
- 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.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.R. Bañuelos, R. G. Smits. Brownian motions in cones. P. T. R. F. 108, 1997, 299–319.zbMATHCrossRefGoogle Scholar
- 5.O. Chybiryakov. Processus de Dunkl et relation de Lamperti. Thèse de doctorat, Université Paris VI, June 2005.Google Scholar
- 6.N. Demni, M. Zani. Large deviations for statistics of Jacobi process. To appear in S. P. A. Google Scholar
- 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.N. Demni. Generalized Bessel function of type D. SIGMA Journal. 4, 2008, 075, 7 pages.Google Scholar
- 9.N. Demni. Note on radial Dunkl processes. Submitted to Ann. I. H. P. Google Scholar
- 10.Y. Doumerc. Matrix Jacobi Process. Thèse de doctorat, Université Paul Sabatier, May 2005.Google Scholar
- 11.C. F. Dunkl, Y. Xu. Orthogonal Polynomials of Several Variables. Encyclopedia of Mathematics and Its Applications. Cambridge University Press. 2001.Google Scholar
- 12.C. F. Dunkl. Differential-difference operators associated to reflection groups. Trans. Amer. Math. Soc. 311. 1989, no. 1, 167–183.MathSciNetzbMATHCrossRefGoogle Scholar
- 13.C. F. Dunkl. Integral kernels with reflection group invariance. Canad. J. Math. 43. 1991, no. 6, 1213–1227.MathSciNetzbMATHCrossRefGoogle Scholar
- 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.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.J. E. Humphreys. Reflections Groups and Coxeter Groups. Cambridge University Press. 29. 2000.Google Scholar
- 17.N. N. Lebedev. Special Functions and their Applications. Dover Publications, INC. 1972.Google Scholar
- 18.D. Revuz, M. Yor. Continuous Martingales and Brownian Motion, 3rd ed., Springer, 1999.Google Scholar
- 19.W. Schoutens. Stochastic Processes and Orthogonal Polynomials. Lecture Notes in Statistics, 146. Springer, 2000.Google Scholar
- 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.J. Warren, M. Yor. The Brownian Burglar: conditioning Brownian motion by its local time process. Sém. Probab. XXXII., 1998, 328–342.MathSciNetGoogle Scholar
- 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