Dirichlet forms for singular diffusion in higher dimensions

We describe singular diffusion in bounded subsets $\mathrm{\Omega }$ of ${\mathbb{R}}^n$ by form methods and characterize the associated operator. We also prove positivity and contractivity of the corresponding semigroup. This results in a description of a stochastic process moving according to classical diffusion in one part of $\mathrm{\Omega }$, where jumps are allowed through the rest of $\mathrm{\Omega }$.