Dirichlet problem associated with Dunkl Laplacian on $W$-invariant open sets

Combining probabilistic and analytic tools from potential theory, we investigate Dirichlet problems associated with the Dunkl Laplacian ${\mathrm{\Delta }}_k$. We establish, under some conditions on the open set $D\subset {\mathbb{R}}^d$, the existence of a unique continuous function $h$ in the closure of $D$, twice differentiable in $D$, such that $${\mathrm{\Delta }}_{k}h=0\text{in}D\text{and}h=f\text{on}\mathrm{\partial }D\mathrm{.}$$ We also give a probabilistic formula characterizing the solution $h$. The function $f$ is assumed to be continuous on the Euclidean boundary $\mathrm{\partial }D$ of $D$.