Bounded harmonic functions for the Heckman--Opdam Laplacian

Elementary symmetric polynomials and martingales for Heckman-Opdam processes

We consider the generators $L_k$ of Heckman-Opdam diffusion processes in the compact and non-compact case in $N$ dimensions for root systems of type $A$ and $B$, with a multiplicity function of the form $k=κk_0$ with some fixed value $k_0$ and a varying constant $κ\in\,[0,\infty[$. Using elementary symmetric functions, we present polynomials which are simultaneous eigenfunctions of the $L_k$ for all $κ\in\,]0,\infty[$. This leads to martingales associated with the Heckman-Opdam diffusions $ (X_{t,1},\ldots,X_{t,N})_{t\ge0}$. As our results extend to the freezing case $κ=\infty$ with a determin…

The Heckman-Opdam Markov processes

We introduce and study the natural counterpart of the Dunkl Markov processes in a negatively curved setting. We give a semimartingale decomposition of the radial part, and some properties of the jumps. We prove also a law of large numbers, a central limit theorem, and the convergence of the normalized process to the Dunkl process. Eventually we describe the asymptotic behavior of the infinite loop as it was done by Anker, Bougerol and Jeulin in the symmetric spaces setting in \cite{ABJ}.

Matrix models for multilevel Heckman-Opdam and multivariate Bessel measures

We study multilevel matrix ensembles at general beta by identifying them with a class of processes defined via the branching rules for multivariate Bessel and Heckman-Opdam hypergeometric functions. For beta = 1, 2, we express the joint multilevel density of the eigenvalues of a generalized beta-Wishart matrix as a multivariate Bessel ensemble, generalizing a result of Dieker-Warren. In the null case, we prove the conjecture of Borodin-Gorin that the joint multilevel density of the beta-Jacobi ensemble is given by a principally specialized Heckman-Opdam measure.

A Central Limit Theorem for Random Walks on the Dual of a Compact Grassmannian

We consider compact Grassmann manifolds $G/K$ over the real, complex or quaternionic numbers whose spherical functions are Heckman-Opdam polynomials of type $BC$. From an explicit integral representation of these polynomials we deduce a sharp Mehler-Heine formula, that is an approximation of the Heckman-Opdam polynomials in terms of Bessel functions, with a precise estimate on the error term. This result is used to derive a central limit theorem for random walks on the semi-lattice parametrizing the dual of $G/K$, which are constructed by successive decompositions of tensor powers of spherical…

关于通过球函数得到自由乘法布朗运动的显式公式

An explicit formula for free multiplicative Brownian motions via spherical functions

通过球函数推导出自由乘法布朗运动的显式公式，并展示其在不同根系中的应用。

大温度下通过狄利克雷过程的矩阵谐波分析

Matrix harmonic analysis at high temperature via the Dirichlet process

本文通过狄利克雷过程研究大温度下随机矩阵的谐波分析，揭示了经验谱分布与测度之间的马尔可夫-克雷因对应关系。