Potential kernels for radial Dunkl Laplacians

TL;DR

This paper establishes two-sided bounds for the Newton and Poisson kernels associated with the $W$-invariant Dunkl Laplacian in complex symmetric spaces, extending classical results and providing new estimates for Dunkl processes and Dyson Brownian motion.

Contribution

It derives explicit two-sided bounds for kernels of Dunkl Laplacians in complex symmetric spaces, including new estimates for Dunkl processes and applications to Dyson Brownian motion.

Findings

Bounds for Dunkl-Poisson kernel in complex symmetric spaces

Bounds for Newton kernel in higher dimensions

Application to Dyson Brownian motion and Weyl chambers

Abstract

We derive two-sided bounds for the Newton and Poisson kernels of the $W$-invariant Dunkl Laplacian in geometric complex case when the multiplicity $k(\alpha)=1$, i.e. for flat complex symmetric spaces. For the invariant Dunkl-Poisson kernel $P^W(x,y)$, the estimates are $$ P^W(x,y)\asymp \frac{P^{{\bf R}^d}(x,y)}{\prod_{\alpha > 0 \ }|x-\sigma_\alpha y|^{2k(\alpha)}},$$ where the $\alpha$'s are the positive roots of a root system acting in ${\bf R}^d$, the $\sigma_\alpha$'s are the corresponding symmetries and $P^{{\bf R}^d}$ is the classical Poisson kernel in ${{\bf R}^d}$. Analogous bounds are proven for the Newton kernel when $d\ge 3$. The same estimates are derived in the rank one direct product case $\mathbb Z_2^N$ and conjectured for general $W$-invariant Dunkl processes. As an application, we get a two-sided bound for the Poisson and Newton kernels of the classical Dyson Brownian motion and of the Brownian motions in any Weyl chamber.