Skew-product representations of multidimensional Dunkl Markov processes
Annales de l'I.H.P. Probabilités et statistiques, Volume 44 (2008) no. 4, pp. 593-611.

In this paper we obtain skew-product representations of the multidimensional Dunkl processes which generalize the skew-product decomposition in dimension 1 obtained in L. Gallardo and M. Yor. Some remarkable properties of the Dunkl martingales. Séminaire de Probabilités XXXIX, 2006. We also study the radial part of the Dunkl process, i.e. the projection of the Dunkl process on a Weyl chamber.

Dans cet article nous obtenons des produits semi-directs des processus de Dunkl multidimensionnels qui généralisent ceux obtenus en dimension 1 dans L. Gallardo and M. Yor. Some remarkable properties of the Dunkl martingales. In Séminaire de Probabilités XXXIX, 2006. Nous étudions les processus de Dunkl radiaux qui sont les projections des processus de Dunkl sur une chambre de Weyl.

DOI: 10.1214/07-AIHP108
Classification: 60J75, 60J25
Keywords: Dunkl processes, Feller processes, Skew-product, Weyl group
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Chybiryakov, Oleksandr. Skew-product representations of multidimensional Dunkl Markov processes. Annales de l'I.H.P. Probabilités et statistiques, Volume 44 (2008) no. 4, pp. 593-611. doi : 10.1214/07-AIHP108. http://archive.numdam.org/articles/10.1214/07-AIHP108/

[1] J.-P. Anker, P. Bougerol and T. Jeulin. The infinite Brownian loop on a symmetric space. Rev. Mat. Iberoamericana 18 (2002) 41-97. | MR | Zbl

[2] P. Biane, P. Bougerol and N. O'Connell. Littelmann paths and Brownian paths. Duke Math. J. 130 (2005) 127-167. | MR | Zbl

[3] O. Chybiryakov. Processus de Dunkl et relation de Lamperti. PhD Thesis, University Paris 6, 2006.

[4] C. F. Dunkl and Y. Xu. Orthogonal Polynomials of Several Variables. Cambridge University Press, Cambridge, 2001. | MR | Zbl

[5] S. N. Ethier and T. G. Kurtz. Markov Processes. Characterization and Convergence. Wiley, New York, 1986. | MR | Zbl

[6] A. R. Galmarino. Representation of an isotropic diffusion as a skew product. Z. Wahrsch. Verw. Gebiete 1 (1962/1963) 359-378. | MR | Zbl

[7] L. Gallardo and M. Yor. An absolute continuity relationship between two multidimensional Dunkl processes. Private communication, 2005.

[8] L. Gallardo and M. Yor. Some new examples of Markov processes which enjoy the time-inversion property. Probab. Theory Related Fields 132 (2005) 150-162. | MR | Zbl

[9] L. Gallardo and M. Yor. A chaotic representation property of the multidimensional Dunkl processes. Ann. Probab. 34 (2006) 1530-1549. | MR | Zbl

[10] L. Gallardo and M. Yor. Some remarkable properties of the Dunkl martingales. Séminaire de Probabilités XXXIX 337-356. Lecture Notes in Math. 1874. Springer, Berlin, 2006. | MR | Zbl

[11] K. Itô and H. P. Mckean Jr. Diffusion Processes and Their Sample Paths. Die Grundlehren der Mathematischen Wissenschaften, Band 125. Academic Press Inc., Publishers, New York, 1965. | MR | Zbl

[12] N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes. North-Holland Publishing Co., Amsterdam, 1981. | MR | Zbl

[13] I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus, 2nd edition. Springer-Verlag, New York, 1991. | MR | Zbl

[14] H. Kunita. Absolute continuity of Markov processes and generators. Nagoya Math. J. 36 (1969) 1-26. | MR | Zbl

[15] S. Lawi. Towards a characterization of Markov processes enjoying the time-inversion property. J. Theoret. Probab. 21 (2008) 144-168. | MR | Zbl

[16] H. P. Mckean Jr. Stochastic Integrals. Academic Press, New York, 1969. | MR | Zbl

[17] E. J. Pauwels and L. C. G. Rogers. Skew-Product Decompositions of Brownian Motions. Geometry of Random Motion (Ithaca, N.Y., 1987), pp. 237-262. Contemp. Math. 73. Amer. Math. Soc., Providence, RI, 1988. | MR | Zbl

[18] M. Rösler. Generalized Hermite polynomials and the heat equation for Dunkl operators. Comm. Math. Phys. 192 (1998) 519-542. | MR | Zbl

[19] M. Rösler. Dunkl operators: theory and applications. Orthogonal Polynomials and Special Functions (Leuven, 2002), pp. 93-135. Lecture Notes in Math. 1817. Springer, Berlin, 2003. | MR | Zbl

[20] M. Rösler and M. Voit. Markov processes related with Dunkl operators. Adv. in Appl. Math. 21 (1998) 575-643. | MR | Zbl

[21] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Springer-Verlag, Berlin, 1999. | MR | Zbl

[22] M. Yor. Exponential Functionals of Brownian Motion and Related Processes. Springer-Verlag, Berlin, 2001. (With an introductory chapter by Hélyette Geman, Chapters 1, 3, 4, 8 translated from the French by Stephen S. Wilson.) | MR | Zbl

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