Upper and lower bounds for Dunkl heat kernel

On $\mathbb R^N$ equipped with a normalized root system $R$, a multiplicity function $k(\alpha) > 0$, and the associated measure $$dw(\mathbf x)=\prod_{\alpha\in R}|\langle \mathbf x,\alpha\rangle|^{k(\alpha)}\, d\mathbf x,$$ let $h_t(\mathbf x,\mathbf y)$ denote the heat kernel of the semigroup generated by the Dunkl Laplace operator $\Delta_k$. Let $d(\mathbf x,\mathbf y)=\min_{\sigma\in G} \| \mathbf x-\sigma(\mathbf y)\|$, where $G$ is the reflection group associated with $R$. We derive the following upper and lower bounds for $h_t(\mathbf x,\mathbf y)$: for all $c_l>1/4$ and $0<c_u<1/4$ there are constants $C_l,C_u>0$ such that $$C_{l}w(B(\mathbf{x},\sqrt{t}))^{-1}e^{-c_{l}\frac{d(\mathbf{x},\mathbf{y})^2}{t}} \Lambda(\mathbf x,\mathbf y,t) \leq h_t(\mathbf{x},\mathbf{y}) \leq C_{u}w(B(\mathbf{x},\sqrt{t}))^{-1}e^{-c_{u}\frac{d(\mathbf{x},\mathbf{y})^2}{t}} \Lambda(\mathbf x,\mathbf y,t),$$ where $\Lambda(\mathbf x,\mathbf y,t)$ can be expressed by means of some rational functions of $\| \mathbf x-\sigma(\mathbf y)\|/\sqrt{t}$. An exact formula for $\Lambda(\mathbf x,\mathbf y,t)$ is provided.