## A chaotic representation property of the multidimensional Dunkl processes.(English)Zbl 1107.60015

A Dunkl Markov process in $$R^d$$ is a càdlàg Markov process with infinitesimal generator ${\mathcal L}_k= {1\over 2}L_k\equiv {1\over 2}\sum^d_{i=1} T^2_i.$ Here, $$T_i$$ $$(1\leq i\leq d)$$ is the differential-difference operator defined for $$u\in C^1(R^d)$$ by $T_i u(x)= {\partial u(x)\over\partial x_i}+ \sum_{\alpha\in R_+} k(\alpha)\alpha_i{u(x)- u(\sigma_\alpha x)\over\langle\alpha, x\rangle}$ ($$\langle\cdot,\cdot\rangle$$ denoting the Euclidean scalar product) where $$R$$ is a root system in $$\mathbb{R}^d$$, $$R_+$$ is a positive subsystem; $$k(\cdot)$$ is a nonnegative multiplicity function defined on $$R$$ and invariant by the finite reflection group $$W$$ associated with $$R$$, and $$\sigma_\alpha$$ is the reflection with respect to the hyperplane $$H_\alpha$$ orthogonal to $$\alpha$$ [see, e.g., C. Dunkl, Trans. Am. Math. Soc. 311, No. 1, 167–183 (1989; Zbl 0652.33004)]. It is assumed that, for all $$\alpha\in R$$, $$\langle\alpha,\alpha\rangle= 2$$, so that $\sigma_\alpha(x)= x-\langle\alpha, x\rangle\alpha,\quad x\in R^d.$ The authors study Dunkl processes $$X$$ in $$\mathbb{R}^d$$ which are also martingales. They obtain the martingale decomposition of $$X$$ into its continuous part and its purely discontinuous part. The main results are (a) a skew-product decomposition of the one-dimendional Dunkl process, and (b) a chaos decomposition of $$X$$.

### MSC:

 60G17 Sample path properties 60G44 Martingales with continuous parameter 60J25 Continuous-time Markov processes on general state spaces 60J60 Diffusion processes 60J65 Brownian motion 60J75 Jump processes (MSC2010) 60H05 Stochastic integrals

Zbl 0652.33004
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