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Radial Dunkl processes associated with dihedral systems, Séminaires de Probabilités
"... Abstract. We give some interest in radial Dunkl processes associated with dihedral systems. We write down the semi group density and as a byproduct the generalized Bessel function and the Winvariant generalized Hermite polynomials. Then, a skew product decomposition, involving only independent Bes ..."
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Abstract. We give some interest in radial Dunkl processes associated with dihedral systems. We write down the semi group density and as a byproduct the generalized Bessel function and the Winvariant 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 2 (C)functions as Lku(x) = 1 ∑ < ∇u(x), α> ∆u(x) + k(α) 2 < x, α> α∈R+ 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 Winvariant function. The semi group density of X with respect to the Lebesgue measure in V is given by (1) p k 1 t (x, y) = cktγ+m/2e−(x2 +y  2 ()/2t W x√t Dk, y) √ ω