Potential kernels for radial Dunkl Laplacians

Graczyk, P. ; Luks, T. ; Sawyer, P.

Canadian journal of mathematics, Tome 74 (2022) no. 4, pp. 1005-1033

Résumé

We derive two-sided bounds for the Newton and Poisson kernels of the W-invariant Dunkl Laplacian in the 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 $$ \begin{align*} P^{W}(x,y)\asymp \frac{P^{\mathbf{R}^{d}}(x,y)}{\prod_{\alpha> 0 \ }|x-\sigma_{\alpha} y|^{2k(\alpha)}}, \end{align*} $$where the $\alpha $’s are the positive roots of a root system acting in $\mathbf {R}^{d}$, the $\sigma _{\alpha }$’s are the corresponding symmetries and $P^{\mathbf {R}^{d}}$ is the classical Poisson kernel in ${\mathbf {R}^{d}}$. 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.

Détail

DOI : 10.4153/S0008414X21000195

Mots-clés : Potential kernel, Newton kernel, Dunkl setting, complex symmetric space, root system