## 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
Full Text:

### References:

 [1] Biane, P. (1990). Chaotic representations for finite Markov chains. Stochastics Stochastic Rep. 30 61–68. · Zbl 0712.60049 [2] Biane, P., Bougerol, P. and O’Connell, N. (2005). Littlemann paths and Brownian paths. Duke Math. J. 130 127–167. · Zbl 1161.60330 [3] Chybiryakov, O. (2005). Skew-product representations of multidimensional Dunkl Markov processes. · Zbl 1180.60072 [4] Dellacherie, C., Maisonneuve, B. and Meyer, P. A. (1992). Probabilités et potentiel . Chapitres XVII à XXIV. Processus de Markov (fin). Compléments de calcul stochastique . Hermann, Paris. [5] Dellacherie, C. and Meyer, P. A. (1987). Probabilités et potentiel . Chapitres XII à XVI. Théorie du potentiel associée à une résolvante. Théorie des processus de Markov . Hermann, Paris. [6] van Diejen, J. F. and Vinet, L. (2000). Calogero–Sutherland–Moser Models . Springer, New York. · Zbl 0942.00063 [7] Dunkl, C. (1989). Differential-differences operators associated to reflection groups. Trans. Amer. Math. Soc. 311 167–183. JSTOR: · Zbl 0652.33004 [8] Dunkl, C. (1992). Hankel transforms associated to finite reflection groups. Contemp. Math. 138 123–138. · Zbl 0789.33008 [9] Dunkl, C. and Xu, Y. (2001). Orthogonal Polynomials of Several Variables . Cambridge Univ. Press. · Zbl 0964.33001 [10] Feller, W. (1971). An Introduction to Probability Theory and Its Applications 2 , 2nd ed. Wiley, New York. · Zbl 0199.32204 [11] Gallardo, L. and Yor, M. (2006). Some remarkable properties of the Dunkl martingales. Séminaire de Probabilités XXXIX . Lecture Notes in Math. 1874 . In Memoriam Paul-Andre Mayer . Springer, Berlin. · Zbl 1128.60027 [12] Gallardo, L. and Yor, M. (2005). Some new examples of Markov processes which enjoy the time-inversion property. Probab. Theory Related Fields 132 150–162. · Zbl 1087.60058 [13] Godefroy, L. (2003). Frontière de Martin sur les hypergroupes et principe d’invariance relative au processus de Dunkl. Thèse, Univ. Tours. [14] Isobe, E. and Sato, S. (1983). Wiener–Hermite expansion of a process generated by an Itô stochastic differential equation. J. Appl. Probab. 20 754–765. JSTOR: · Zbl 0528.60055 [15] Lamperti, J. (1972). Semi-stable Markov processes I. Z. Wahrsch. Verw. Gebiete 22 205–225. · Zbl 0274.60052 [16] Meyer, P. A. (1967). Intégrales stochastiques, I–IV. Séminaire de Probabilités I . Lecture Notes in Math. 39 72–162. Springer, Berlin. · Zbl 0157.25001 [17] Meyer, P. A. (1989). Construction de solutions d’équations de structure. Séminaire de Probabilités XXIII . Lecture Notes in Math. 1372 142–145. Springer, Berlin. · Zbl 0739.60050 [18] Rösler, M. (1999). Generalized Hermite polynomials and the heat equation for Dunkl operators. Comm. Math. Phys. 192 519–542. · Zbl 0908.33005 [19] Rösler, M. (2003). Dunkl operators: Theory and applications. Orthogonal Polynomials and Special Functions . Lecture Notes in Math. 1817 93–136. Springer, Berlin. · Zbl 1029.43001 [20] Rösler, M. and Voit, M. (1998). Markov processes related with Dunkl operators. Adv. in Appl. Math. 21 575–643. · Zbl 0919.60072 [21] Trimèche, K. (2002). The Dunkl intertwining operator on spaces of functions and distributions and integral representation of its dual. Integral Transform. Spec. Funct. 12 349–374. · Zbl 1027.47027 [22] Watanabe, S. (1964). On discontinuous additive functionals and Lévy measures of a Markov process. Japan. J. Math. 34 53–79. · Zbl 0141.15703 [23] Yor, M. (1997). Some Aspects of Brownian Motion. Part II: Some Recent Martingale Problems . Birkhäuser, Basel. · Zbl 0880.60082
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.