SIGMA 4 (2008), 074, 14 pages      arXiv:0811.0504      https://doi.org/10.3842/SIGMA.2008.074
Contribution to the Special Issue on Dunkl Operators and Related Topics

First Hitting Time of the Boundary of the Weyl Chamber by Radial Dunkl Processes

Nizar Demni
SFB 701, Fakultät für Mathematik, Universität Bielefeld, Deutschland

Received July 01, 2008, in final form October 24, 2008; Published online November 04, 2008

Abstract
We provide two equivalent approaches for computing the tail distribution of the first hitting time of the boundary of the Weyl chamber by a radial Dunkl process. The first approach is based on a spectral problem with initial value. The second one expresses the tail distribution by means of the W-invariant Dunkl-Hermite polynomials. Illustrative examples are given by the irreducible root systems of types A, B, D. The paper ends with an interest in the case of Brownian motions for which our formulae take determinantal forms.

Key words: radial Dunkl processes; Weyl chambers; hitting time; multivariate special functions; generalized Hermite polynomials.

pdf (273 kb)   ps (193 kb)   tex (15 kb)

References

1. Baker T.H., Forrester P.J., The Calogero-Sutherland model and generalized classical polynomials, Comm. Math. Phys. 188 (1997), 175-216, solv-int/9608004.
2. Bañuelos R., Smits R.G., Brownian motions in cones, Probab. Theory Related Fields 108 (1997), 299-319.
3. Beerends R.J., Opdam E.M., Certain hypergeometric series related to the root system BC, Trans. Amer. Math. Soc. 339 (1993), 581-607.
4. Chybiryakov O., Processus de Dunkl et Relation de Lamperti, Ph.D. Thesis, Paris VI University, June 2005.
5. Comtet A., Desbois J., Brownian motion in wedges, last passage time and the second arc-sine law, J. Phys. A: Math. Gen. 36 (2003), no. 17, L255-L261, cond-mat/0302269.
6. Demni N., Generalized Bessel function of type D, SIGMA 4 (2007), 075, 7 pages, arXiv:0811.0507.
7. Demni N., Radial Dunkl processes associated with dihedral systems, Séminaires de Probabilités, to appear, arXiv:0707.0367.
8. Doumerc Y., O'Connell N., Exit problems associated with finite reflection groups, Probab. Theory Related Fields 132 (2005), 501-538, arXiv:0707.2009.
9. Dunkl C.F., Xu Y., Orthogonal polynomials of several variables, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2001.
10. Dunkl C.F., Intertwining operators associated to the group S₃, Trans. Amer. Math. Soc. 347 (1995), 3347-3374.
11. Gallardo L., Yor M., Some new examples of Markov processes which enjoy the time-inversion property, Probab. Theory Related Fields 132 (2005), 150-162.
12. Gallardo L., Yor M., A chaotic representation property of the multidimensional Dunkl processes, Ann. Probab. 34 (2006), 1530-1549, math.PR/0609679.
13. Gallardo L., Yor M., Some remarkable properties of the Dunkl martingales, in Memoriam Paul-André Meyer: Séminaire de Probabilités XXXIX, Lecture Notes in Math., Vol. 1874, Springer, Berlin, 2006, 337-356.
14. Gallardo L., Godefroy L., An invariance principle related to a process which generalizes N-dimensional Brownian motion, C. R. Math. Acad. Sci. Paris 338 (2004), 487-492.
15. Humphreys J.E., Reflections groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, Vol. 29, Cambridge University Press, Cambridge, 1990.
16. Lassalle M., Polynômes de Laguerre généralisés, C. R. A. S. Paris Série I 312 (1991), 725-728.
17. Lassalle M., Polynômes d'Hermite généralisés, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), 725-728.
18. Lebedev N.N., Special functions and their applications, Dover Publications, Inc., New York, 1972.
19. Rösler M., Voit M., Markov processes related with Dunkl operators, Adv. in Appl. Math. 21 (1998), 575-643.
20. Rösler M., Dunkl operator: theory and applications, in Orthogonal Polynomials and Special Functions (Leuven, 2002), Lecture Notes in Math., Vol. 1817, Springer, Berlin, 2003, 93-135, math.CA/0210366.
21. Stembridge J.R., Nonintersecting paths, Pfaffians and plane partitions, Adv. Math. 83 (1990), 96-131.