Elementary symmetric polynomials and martingales for Heckman-Opdam processes

TL;DR

This paper develops elementary symmetric polynomial eigenfunctions for Heckman-Opdam diffusion processes, establishing associated martingales and extending results to the deterministic limit as the multiplicity parameter grows large.

Contribution

It introduces polynomial eigenfunctions for Heckman-Opdam generators using elementary symmetric functions, enabling new martingale constructions and analysis of the freezing limit.

Findings

Eigenfunctions of Heckman-Opdam generators are expressed via elementary symmetric functions.

Martingales are constructed for the associated diffusion processes.

Formulas for expectations in the freezing limit are derived.

Abstract

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 $k=\kappa k_0$ with some fixed value $k_0$ and a varying constant $\kappa\in\,[0,\infty[$. Using elementary symmetric functions, we present polynomials which are simultaneous eigenfunctions of the $L_k$ for all $\kappa\in\,]0,\infty[$. 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 $\kappa=\infty$ with a deterministic limit after some renormalization, we find formulas for the expectations $\mathbb E(\prod_{j=1}^N(y-X_{t,j})),$ $y\in\mathbb C$.