Abstract
We introduce and study the natural counterpart of the Dunkl Markov processes in a negatively curved setting. We give a semimartingale decomposition of the radial part, and some properties of the jumps. We prove also a law of large numbers, a central limit theorem, and the convergence of the normalized process to the Dunkl process. Eventually we describe the asymptotic behavior of the infinite loop as it was done by Anker, Bougerol and Jeulin in the symmetric spaces setting in (Iberoamericana 18: 41–97, 2002).
References
Anker J-Ph., Bougerol Ph., Jeulin T., (2002). The infinite Brownian loop on a symmetric space. Rev. Mat. Iberoamericana 18:41–97
Babillot M., (1994). A probabilistic approach to heat diffusion on symmetric spaces. J. Theor. Prob 7:599–607
Biane Ph., (1994). Quelques propriétés du mouvement Brownien dans un cône. Stoch. Process. Appl 53:233–240
Biane Ph., Bougerol Ph., O’Connell N., (2005). Littelmann paths and Brownian paths. Duke Math. J. 130:127–167
Bougerol Ph., Jeulin T., (2001). Brownian bridge on Riemannian symmetric spaces. C. R. Acad. Sci. Paris Sér. I Math. 333:785–790
Bougerol Ph., Jeulin T., (1999). Brownian bridge on hyperbolic spaces and on homogeneous trees. Probab. Th. Rel. Fields 115:95–120
Bourbaki, N.: Groupes et algèbres de Lie Chap. 4–6. Hermann, Paris, (1968); Masson, Paris, (1981)
Cépa E., (1995). Equations différentielles stochastiques multivoques. Sém. Probab. XXIX:86–107
Cépa E., Lépingle D., (2001). Brownian particles with electrostatic repulsion on the circle : Dyson’s model for unitary random matrices revisited. ESAIM Probab. Statist. 5:203–224
Cherednik I., (1991). A unification of Knizhnik-Zamolodchnikov equations and Dunkl operators via affine Hecke algebras. Invent. Math. 106:411–432
Chybiryakov, O.: Skew-product representations of multidimensional Dunkl Markov processes (submitted)
Chybiryakov, O.: Radial Dunkl Markov processes, local times and a multidimensional extension of Lévy’s equivalence (in preparation)
Dubédat J., (2004). Reflected planar Brownian motions, intertwining relations and crossing probabilities. Ann. Inst. H. Poincaré Prob. Stat. 40:539–552
Ethier, N., Kurtz, G.: Markov processes. Characterization and convergence. Wiley Series Prob. Math. Stat. (1986)
Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet forms and symmetric Markov processes. de Gruyter Stud. Math. 19, (1994)
Gallardo, L. Yor, M. Some remarkable properties of the Dunkl martingales. Sém. Probab. XXXVIII (in press). Dedicated to Paul André Meyer, Springer (2005)
Gallardo L., Yor M., (2005). Some new examples of Markov processes which enjoys the time-inversion property. Prob. Theory Relat. Fields 132:150–162
Gallardo, L., Yor, M.: A chaotic representation property of the multidimensional Dunkl processes. Ann. Prob. (in press)
Heckman G.J., Opdam E.M., (1987). Root systems and hypergeometric functions I. Compositio Math. 64:329–352
Helgason, S.: Groups and Geometric Analysis. Academic Press (1984)
Jacod J., (1979). Calcul stochastique et problèmes de martingales. Lecture Notes in Mathematics, vol. 714. Springer, Berlin Heidelberg New York
Jacod J., Shiryaev A.N., (1987). Limit Theorems for Stochastic Processes. Springer, Berlin Heidelberg New York
De Jeu M.F.E., (1993). The Dunkl transform. Invent. Math. 113:147–162
De Jeu, M.F.E.: Paley-Wiener theorems for the Dunkl transform. Available on arxiv math.CA/0404439
Lawler G.F., Schramm O., Werner W., (2003). Conformal restriction: the chordal case. J. Amer. Math. Soc. 16:917–955
Le Gall J-F., (1987). Mouvement brownien, cônes et processus stables (French). Prob. Theory Relat. Fields 76:587–627
Meyer P.A. (1967) Intégrales stochastiques. Sém. Prob. I, Lect. Notes Math. vol. 39. Springer, Berlin Heidelberg New York
O’Connell N., (2003). Random matrices, non-colliding processes and queues, Sém. Probab. XXXVI, pp. 165–182 Lecture Notes in Math. vol. 1801. Springer, Berlin Heidelberg New York
Opdam E.M., (1995). Harmonic analysis for certain representations of graded Hecke algebras. Acta. Math. 175:75–121
Revuz D., Yor M. (1999) Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin Heidelberg New York (1999)
Rösler M., (1998). Generalized Hermite polynomials and the heat equation for Dunkl operators. Comm. Math. Phys. 192(3):519–542
Rösler M., Voit M., (1998). Markov processes related with Dunkl operators. Adv. Appl. Math. 21(4):575–643
Schapira, Br.: Contributions to the hypergeometric function theory of Heckman and Opdam: sharp estimates, Schwartz space, heat kernel. Available on arxiv math.CA/0605045
Schapira, Br.: Marches aléatoires sur un immeuble affine de type à r et mouvement brownien de la chambre de Weyl (French) (in preparation)
Author information
Authors and Affiliations
Corresponding author
Additional information
Partially supported by the European Commission (IHP Network HARP 2002–2006).
Rights and permissions
About this article
Cite this article
Schapira, B. The Heckman–Opdam Markov processes. Probab. Theory Relat. Fields 138, 495–519 (2007). https://doi.org/10.1007/s00440-006-0034-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-006-0034-1
Keywords
- Markov processes
- Jump processes
- Root systems
- Dirichlet forms
- Dunkl processes
- Limit theorems
Mathematical Subject Classification (2000)
- 58J65
- 60B15
- 60F05
- 60F17
- 60J35
- 60J60
- 60J65
- 60J75