Affine Dunkl processes of type ${\tilde{\mathrm{A}}}_1$

Chapon, François

Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 3, pp. 854-870.

Nous introduisons l’analogue des processus de Dunkl dans le cas d’un système de racines affines de type ${\tilde{A}}_1$. La construction du processus de Dunkl affine est obtenue par une décomposition en skew-product de sa partie radiale et d’un processus de sauts sur le groupe de Weyl affine, la partie radiale du processus de Dunkl affine étant définie par un processus gaussien sur l’hypergroupe ultrasphérique $[0,1]$. Nous montrons que le processus de Dunkl affine est un processus de Markov càdlàg ainsi qu’une martingale locale, étudions ses sauts, et donnons sa décomposition en martingale, propriétés analogues à celles du processus de Dunkl classique.

We introduce the analogue of Dunkl processes in the case of an affine root system of type ${\tilde{A}}_1$. The construction of the affine Dunkl process is achieved by a skew-product decomposition by means of its radial part and a jump process on the affine Weyl group, where the radial part of the affine Dunkl process is given by a Gaussian process on the ultraspherical hypergroup $[0,1]$. We prove that the affine Dunkl process is a càdlàg Markov process as well as a local martingale, study its jumps, and give a martingale decomposition, which are properties similar to those of the classical Dunkl process.

[1] M. Aigner and G. M. Ziegler. La fonction cotangente et l'astuce de Herglotz, ch. 20. In Raisonnements Divins. Springer, Paris, 2006.

[2] R. Askey. Orthogonal Polynomials and Special Functions. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1975. | MR | Zbl

[3] D. Bakry and N. Huet. The hypergroup property and representation of Markov kernels. In Séminaire de Probabilités XLI 295-347. Lecture Notes in Math. 1934. Springer, Berlin, 2008. | MR | Zbl

[4] P. Biane. Matrix valued Brownian motion and a paper by Pólya. In Séminaire de Probabilités XLII 171-185. Lecture Notes in Math. 1979. Springer, Berlin, 2009. | MR | Zbl

[5] W. R. Bloom and H. Heyer. Harmonic Analysis of Probability Measures on Hypergroups. de Gruyter Studies in Mathematics 20. de Gruyter, Berlin, 1995. | MR | Zbl

[6] S. Bochner. Sturm-Liouville and heat equations whose eigenfunctions are ultraspherical polynomials or associated Bessel functions. In Proceedings of the Conference on Differential Equations (Dedicated to A. Weinstein) 23-48. Univ. Maryland Book Store, College Park, MD, 1956. | MR | Zbl

[7] O. Chybiryakov. Skew-product representations of multidimensional Dunkl Markov processes. Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008) 593-611. | Numdam | MR | Zbl

[8] O. Chybiryakov, N. Demni, L. Gallardo, M. Rösler, M. Voit and M. Yor. Harmonic & Stochastic Analysis of Dunkl Processes. Hermann, Paris, 2008.

[9] N. Demni. Radial Dunkl processes associated with dihedral systems. In Séminaire de Probabilités XLII 153-169. Lecture Notes in Math. 1979. Springer, Berlin, 2009. | MR | Zbl

[10] N. Demni. Radial Dunkl processes: Existence, uniqueness and hitting time. C. R. Math. Acad. Sci. Paris 347 (2009) 1125-1128. | MR | Zbl

[11] C. F. Dunkl. Differential-difference operators associated to reflection groups. Trans. Amer. Math. Soc. 311 (1989) 167-183. | MR | Zbl

[12] E. B. Dynkin. Markov Processes, Vols I, II. Academic Press, New York, 1965. | MR

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

[14] 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

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

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

[17] G. Gasper. Banach algebras for Jacobi series and positivity of a kernel. Ann. of Math. (2) 95 (1972) 261-280. | MR | Zbl

[18] J. E. Humphreys. Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics 29. Cambridge Univ. Press, Cambridge, 1990. | MR | Zbl

[19] I. Karaztas and S. E. Shreve. Brownian Motion and Stochastic Calculus, 2nd edition. Springer, New York, 1991. | MR | Zbl

[20] S. Karlin and J. Mcgregor. Classical diffusion processes and total positivity. J. Math. Anal. Appl. 1 (1960) 163-183. | MR | Zbl

[21] S. Karlin and H. M. Taylor. A Second Course in Stochastic Processes. Academic Press, New York, 1981. | MR | Zbl

[22] W. Magnus, F. Oberhettinger and R. P. Soni. Formulas and Theorems for the Special Functions of Mathematical Physics. Springer, New York, 1966. | MR | Zbl

[23] P.-A. Meyer. Intégrales stochastiques IV. In Séminaire de Probabilités (Univ. Strasbourg, Strasbourg, 1966/67), Vol. I 142-162. Springer, Berlin, 1967. | Numdam | MR | Zbl

[24] C. Rentzsch and M. Voit. Lévy processes on commutative hypergroups. In Probability on Algebraic Structures (Gainesville, FL, 1999) 83-105. Contemp. Math. 261. Amer. Math. Soc., Providence, RI, 2000. | MR | Zbl

[25] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. Springer, Berlin, 1999. | MR | Zbl

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

[27] B. Schapira. The Heckman-Opdam Markov processes. Probab. Theory Related Fields 138 (2007) 495-519. | MR | Zbl

[28] B. Schapira. Bounded harmonic functions for the Heckman-Opdam Laplacian. Int. Math. Res. Not. IMRN (2009) 3149-3159. | MR | Zbl

[29] G. Szegő. Orthogonal Polynomials. Amer. Math. Soc., Providence, RI, 1975.

[30] M. Voit. Asymptotc behavior of heat kernels on spheres of large dimensions. J. Multivariate Anal. 59 (1996) 230-248. | MR | Zbl

[31] M. Voit. Rate of convergence to Gaussian measures on $n$-spheres and Jacobi hypergroups. Ann. Probab. 25 (1997) 457-477. | MR | Zbl