DUNKL OPERATORS: PROBABILISTIC OVERVIEW NIZAR DEMNI 1

Abstract. This survey may be considered as a continuation of the one written by M. Rösler on the theory of Dunkl operators. However, we focus on the probabilistic progress on this topic during approximately the last five years. We start with the study of the non symmetric case, that is, the right-continuous with left-hand limits Dunkl processes valued in a finite dimensional euclidean space V: time inversion property, two skew product decompositions, the analysis of jumps, martingale representation and Itô’s formula, the Wiener Chaotic decomposition and an invariance principle. Some interesting consequences in the rank-one case are given, among which we cite the characterization of the Dunkl process using Gilat’s Theorem. Then, we move to the symmetric or the so-called radial part: explicit expressions of the symmetrized Dunkl kernels are written down, a SDE with a singular drift is satisfied, absolute-continuity relations are set and precise results on the first hitting time of the boundary of the positive Weyl chamber are derived. The survey ends with some related processes such as Heckman-Opdam processes and eigenvalues of some matrix-valued processes as well as open questions. 1