Note on Radial Dunkl Processes

This note encloses relatively short proofs of the following known results: the radial Dunkl process associated with a reduced system and a strictly positive multiplicity function is the unique strong solution for all time t of a stochastic differential equation of a singular drift (see [12] for the original proof and [4] for a proof under additional restrictions), the first hitting time of the Weyl chamber by a radial Dunkl process is finite almost surely for small values of the multiplicity function. Our proof of the second mentioned result gives more information than the original one. 1. Preliminaries To be self-contained, we begin by pointing out some facts on root systems and radial Dunkl processes. We refer to [11] for the Dunkl theory, to [8] for a background on root systems and [4] and references therein for facts on radial Dunkl processes. Let (V, 〈, 〉) be a finite real Euclidean space of dimension m. A reduced root system R is a finite set of non zero vectors in V such that : 1 R ∩ Rα = {α,−α} for all α ∈ R, 2 σα(R) = R, where σα is the reflection with respect to the hyperplane Hα orthogonal to α: σα(x) = x− 2 〈α, x〉 |α|2 α, |α| 2 := 〈α, α〉 x ∈ V. A simple system S is a basis of span(R) which induces a total ordering in R. A root α is positive if it is a positive linear combination of elements of S. The set of positive roots is called a positive system and is denoted by R+. The (finite) reflection group W is the group generated by all the reflections σα for α ∈ R. Given a root system R with positive and simple systems R+, S, define the positive Weyl chamber C by: C := {x ∈ V, 〈α, x〉 > 0 ∀α ∈ R+} = {x ∈ V, 〈α, x〉 > 0 ∀α ∈ S} and C, ∂C its closure and boundary respectively. The radial Dunkl process X is defined as the C-valued continuous paths Markov process whose generator is given Date: December 26, 2008. SFB, Bielefeld university, e-mail: [email protected]