MARKOV-PROCESSES RELATED WITH DUNKL OPERATORS

Dunkl operators are parameterized differential-difference operators on R-N that are related to finite reflection groups. They can be regarde d as a generalization of partial derivatives and play a major role in the study of Calogero-Moser-Sutherland-type quantum many-body systems. Dunkl operators lead to generalizations of various analytic structure s, like the Laplace operator, the Fourier transform, Hermite polynomia ls, and the heat semigroup. In this paper we investigate some probabil istic aspects of this theory in a systematic way. For this, we introdu ce a concept of homogeneity of Markov processes on R-N that generalize s the classical notion of processes with independent, stationary incre ments to the Dunkl setting. This includes analogues of Brownian motion and Cauchy processes. The generalizations of Brownian motion have the cadlag property and form, after symmetrization with respect to the un derlying reflection groups, diffusions on the Weyl chambers. A major p art of the paper is devoted to the concept of modified moments of prob ability measures on R-N in the Dunkl setting. This leads to several re sults for homogeneous Markov processes (in our extended setting), incl uding martingale characterizations and limit theorems. Furthermore, re lations to generalized Hermite polynomials, Appell systems, and Ornste in-Uhlenbeck processes are discussed. (C) 1998 Academic Press