Radial Dunkl processes: existence, uniqueness and hitting time

Abstract. This paper is essentially concerned with the study of the radial Dunkl process associated with a reduced root system. It is shown that this diffusion is the unique strong solution for all t ≥ 0 of a stochastic differentiel equation (SDE) with singular drift. Then we prove, via stochastic calculus, that the first hitting time T0 of the positive Weyl chamber is finite for small values of the multiplicity function. Once we did, we compute the tail distribution of T0 and we highlight how radial Dunkl processes generalize the eigenvalues of well known selfadjoint matrix-valued processes which is twofold. On the one hand, existence and uniqueness results for the eigenvalue process of some matrix diffusions are improved. On the other hand, the tail distributions of the first collision time and the first time the smallest eigenvalue hits zero are recovered. Next, we use determinantal representations of multivariate special functions to recover results by Grabiner on Brownian motions in Weyl chambers, to write down the generalized Bessel function for theD-type root system and to deal with the particular case of the Brownian motion reflected on the boundary of the Weyl chamber. Finally, we investigate the trigonometric analog which involves the non-reduced root system of type BC and is related to the eigenvalues processes of the real and complex matrix Jacobi processes. Our existence and uniqueness result is extended to this setting and the semi-group density is given. 1. Preliminaries We begin by pointing out some facts on root systems and radial Dunkl processes. We refer to [36] for the Dunkl theory, to both [7] and [26] for a background on root systems and [11] 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