Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

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.

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