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.

我们在多重性$k(\alpha)=1$ 即对于平坦复对称空间的几何复数情况下推导出W不变Dunkl Laplacian的牛顿核和泊松核的双边界。对于不变的 Dunkl–Poisson 核 $P^{W}(x,y)$ ，估计值为 $$ \begin{align*} P^{W}(x,y)\asymp \frac{P^{\ mathbf{R}^{d}}(x,y)}{\prod_{\alpha> 0 \ }|x-\sigma_{\alpha} y|^{2k(\alpha)}}, \end{align *} $$ 其中 $\alpha $ 是在 $\mathbf {R}^{d}$ 中起作用的根系统的正根，$ \sigma _{\alpha }$ 是相应的对称性 $P^{\mathbf {R}^{d}}$ 是经典的泊松核 ${\mathbf {R}^{d}}$ 。 当$d\ge 3$ 时，牛顿核也证明了类似的界限。

相同的估计是在第一级直接积案例 $\mathbb {Z}_{2}^{N}$ 中得出的，并推测为一般W不变的 Dunkl 过程。

作为一个应用，我们得到了经典戴森布朗运动和任何外尔室中布朗运动的泊松和牛顿核的双边界。