Affine Dunkl processes of type A ˜ 1

François Chapon

Annales de l'I.H.P. Probabilités et statistiques (2012)

  • Volume: 48, Issue: 3, page 854-870
  • ISSN: 0246-0203

Abstract

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We introduce the analogue of Dunkl processes in the case of an affine root system of type 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.

How to cite

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Chapon, François. "Affine Dunkl processes of type $\widetilde{\mathrm {A}}_{1}$." Annales de l'I.H.P. Probabilités et statistiques 48.3 (2012): 854-870. <http://eudml.org/doc/272012>.

@article{Chapon2012,
abstract = {We introduce the analogue of Dunkl processes in the case of an affine root system of type $\widetilde\{\operatorname\{A\}\}_\{1\}$. 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.},
author = {Chapon, François},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Dunkl processes; diffusion processes; orthogonal polynomials; Skew-product decomposition; affine root system; Weyl group; ultraspherical polynomials; Heckman-Opdam processes; affine root systems; Weyl groups; martingale decomposition},
language = {eng},
number = {3},
pages = {854-870},
publisher = {Gauthier-Villars},
title = {Affine Dunkl processes of type $\widetilde\{\mathrm \{A\}\}_\{1\}$},
url = {http://eudml.org/doc/272012},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Chapon, François
TI - Affine Dunkl processes of type $\widetilde{\mathrm {A}}_{1}$
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2012
PB - Gauthier-Villars
VL - 48
IS - 3
SP - 854
EP - 870
AB - We introduce the analogue of Dunkl processes in the case of an affine root system of type $\widetilde{\operatorname{A}}_{1}$. 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.
LA - eng
KW - Dunkl processes; diffusion processes; orthogonal polynomials; Skew-product decomposition; affine root system; Weyl group; ultraspherical polynomials; Heckman-Opdam processes; affine root systems; Weyl groups; martingale decomposition
UR - http://eudml.org/doc/272012
ER -

References

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