Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### Some remarkable properties of the Dunkl martingales

Auteur(s):

Code(s) de Classification MSC:

• 60G17 Sample path properties
• 60G44 Martingales with continuous parameter
• 60J25 Markov processes with continuous parameter
• 60H05 Stochastic integrals

Résumé: In this paper, we study a class, depending on a parameter ${k\geq0}$, of real Feller processes $X^{(k)}=(X^{(k)}_{t})_{t>0}$, the so called Dunkl processes which are martingales satisfying the brownian scaling property. These processes are the only martingales whose absolute value is a Bessel process. Moreover, the absolute continuity and intertwining relations valid for some pairs of Bessel processes may be generalized to Dunkl processes. The main result of the paper is a mixed chaotic representation property for the $L^{2}$ space of the martingale $X^{(k)}$ in terms of its continuous part (which is a brownian motion) and its purely discontinuous part, a martingale $\gamma$ with bracket $<\gamma >_{t} =2kt$.

Mots Clés: Dunkl operator ; Dunkl processes ; Bessel processes ; intertwining semigroups ; Wiener's chaos decomposition

Date: 2004-01-28

Prépublication numéro: PMA-879