Radial Dunkl Processes Associated with Dihedral Systems

We give some interest 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. 1. A quick reminder We refer the reader to [11] and [16] for facts on root systems and to [5], [20] for facts on radial Dunkl processes. Let R be a reduced root system in a finite euclidean space (V,<,>) with positive system R+ and simple system S. Let W be its reflection group and C be its positive Weyl chamber. The radial Dunkl process X associated with R is a continuous paths Markov process valued in C whose generator acts on C(C)-functions as Lku(x) = 1 2 ∆u(x) + ∑ α∈R+ k(α) < ∇u(x), α > < x, α > with < ∇u(x), α >= 0 whenever < x, α >= 0, where ∆,∇ denote the euclidean Laplacian and the gradient respectively and k is a positive multiplicity function, that is, a R+-valued W -invariant function. The semi group density of X with respect to the Lebesgue measure in V is given by (1) pt (x, y) = 1 ckt e 2+|y|2)/2tDW k ( x √ t , y √ t ) ω k(y), x, y ∈ C where γ = ∑ α∈R+ k(α) and m = dimV is the rank of R. The weight function ωk is given by ωk(y) = ∏ α∈R+ < α, y > and D k is the generalized Bessel function. Thus, Lk may be written as (2) Lku(x) = 1 2 ∆u(x)+ < ∇u(x),∇ logωk(x) > . Date: December 29, 2008.