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.

Classification : 60J75, 60J60, 60B15, 33C52
Mots clés : Dunkl processes, diffusion processes, orthogonal polynomials, Skew-product decomposition, affine root system, Weyl group

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