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

Gallardo, Léonard; Yor, Marc

Ann. Probab. 34, No. 4, 1530-1549 (2006).

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 \(X\).

Reviewer: Klaus Schürger (Bonn)

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

Citations:

Zbl 0652.33004

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