In this paper we give a sufficient condition on the semi group densities of an homogeneous Markov process taking values in R n which ensures that it enjoys the time-inversion property. Our condition covers all previously known examples of Markov processes satisfying this property. As new examples we present a class of Markov processes with jumps, the Dunkl processes and their radial parts.

Dunkl processes are martingales as well as càdlàg homogeneous Markov processes taking values in R d and they are naturally associated with a root system. In this paper we study the jumps of these processes, we describe precisely their martingale decompositions into continuous and purely discontinuous parts and we obtain a Wiener chaos decomposition of the corresponding L 2 spaces of these processes in terms of adequate mixed multiple stochastic integrals.

For a root system in R d furnished with its Coxeter-Weyl group W and a multiplicity nonnegative function k, we consider the associated commuting system of Dunkl operators D 1 , . . . , D d and the Dunkl-Laplacian Δ k = D 2 1 +. . .+D 2 d . This paper studies the properties of the functions u defined on an open W -invariant set Ω ⊂ R d and satisfying Δ k u = 0 on Ω (D-harmonicity).In particular, we introduce and give a complete study of a new mean value operator which characterizes D-harmonicity. As applications we prove a strong maximum principle, a Harnack's type theorem and a Bôcher's theorem for D-harmonic functions.