发表机构
Institut für Mathematik, TU Clausthal; Fakultät für Mathematik, Technische Universität Dortmund（Clausthal工业大学数学研究所; 多特蒙德工业大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文介绍有理Dunkl算子及相关马尔可夫过程理论，强调与欧几里得对称空间和Bessel超群的联系，并讨论乘积公式、超群结构及以Dunkl拉普拉斯为生成元的扩散-反射过程。
AI 中文摘要
这些讲义旨在作为有理Dunkl算子、相关特殊函数及相关马尔可夫过程理论的入门介绍，重点强调与欧几里得型黎曼对称空间及正半定矩阵矩阵锥上的Bessel超群相关的例子。我们首先对Dunkl理论进行全面介绍：Dunkl算子、交织算子及其正性、Dunkl核与Dunkl变换、Dunkl拉普拉斯算子及相关热半群。我们进一步概述了与Calogero-Moser-Sutherland模型和广义Hermite多项式的联系。此外，核心关注点将是乘积公式、广义平移以及闭Weyl腔上的相关交换超群结构。特别地，我们解释了特定重数下的Dunkl理论如何与欧几里得型黎曼对称空间及矩阵锥上的Bessel超群相关联，以及这如何导致一系列重数，使得Weyl群不变的Dunkl理论允许保概率平移及闭Weyl腔上的相关交换超群结构。我们最后讨论了与Dunkl理论相关的R^N上的马尔可夫过程，重点强调与群和超群上随机游走的联系。特别地，研究了相关的鞅、鞅刻画、矩函数和Appell特征，即以Dunkl拉普拉斯算子为生成元的扩散-反射过程。
英文摘要
These lecture notes are intended as an introduction to the theory of rational Dunkl operators, the associated special functions and related Markov processes with an emphasis on examples which are related to Riemannian symmetric spaces of Euclidean type and Bessel hypergroups on the matrix cones of positive semidefinite matrices. We start with a comprehensive introduction into Dunkl theory: Dunkl operators, the intertwining operator and its positivity, the Dunkl kernel and the Dunkl transform, the Dunkl Laplacian and the associated heat semigroup. We further give an outline of the connection with Calogero-Moser-Sutherland models and generalized Hermite polynomials. Moreover, of central interest will be product formulas, generalized translations and associated commutative hypergroup structures on closed Weyl chambers. In particular, we explain how Dunkl theory for particular multiplicities is related to Riemannian symmetric spaces of Euclidean type and Bessel hypergroups on the matrix cones, and how this leads to a bunch of multiplicities for which the Weyl-group invariant Dunkl theory admits a probability preserving translation and an associated commutative hypergroup structure on the closed Weyl chamber. We finally discuss Markov processes on R^N which are related with Dunkl theory with an emphasis on connections to random walk on groups and hypergroups. In particular associated martingales, martingale characterizations, moment functions and Appell characters are studied, i.e. diffusion-reflection processes with the Dunkl Laplacians as generators.
CommentsThis is the preprint version of a contribution published in 2008, in a volume based on the Workshop on Stochastic and Harmonic Analysis of Processes with Jumps at Angers, France, 2006. Eds: P. Graczyk et al
Journal refIn: Harmonic and stochastic analysis of Dunkl processes. Eds. P. Graczyk, M. Rösler, M. Yor; Travaux en cours 71, pp. 1-112, Hermann, Paris, 2008
