Affine Dunkl processes of type (A)over-tilde(1)

作者：Chapon Francois^*

来源：Annales de l Institut Henri Poincare-Probabilites et Statistiques, 2012, 48(3): 854-870.

DOI：10.1214/11-AIHP430

• Dunkl processes
• Diffusion processes
• Orthogonal polynomials
• Skew-product decomposition
• Affine root system
• Weyl group

We introduce the analogue of Dunkl processes in the case of an affine root system of type (A) over tilde (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 cadlag 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.