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\begin{document}${{\overline{{G}}\subset {{\mathbb R}}^d}}$\end{document} be a compact set with interior G. Let ρ∈L1(G,dx), ρ>0 dx-a.e. on G, and m:=ρdx. Let A=(aij) be symmetric, and globally uniformly strictly elliptic on G. Let ρ be such that \documentclass[12pt]{minimal}
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\begin{document}${{{{{{\mathcal E}}}}^r(f,g)=\frac{{1}}{{2}}\sum_{{i,j=1}}^{{d}}\int_G a_{{ij}}\partial_i f \partial_j g\,dm}}$\end{document}; f, \documentclass[12pt]{minimal}
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\begin{document}${{g\in C^{{\infty}}(\overline{{G}})}}$\end{document}, is closable in L2(G,m) with closure (ℰr,D(ℰr)). The latter is fulfilled if ρ satisfies the Hamza type condition, or ∂iρ∈L1loc(G,dx), 1≤i≤d. Conservative, non-symmetric diffusion processes Xt related to the extension of a generalized Dirichlet form \documentclass[12pt]{minimal}
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\begin{document}$${{ {{{{\mathcal E}}}}^r(f,g) -\sum_{{i=1}}^{{d}}\int_G \rho^{{-1}}\overline{{B}}_i\partial_i f\, g\, dm; f,g\in D({{{{\mathcal E}}}}^r)_b }}$$\end{document} where \documentclass[12pt]{minimal}
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\begin{document}${{\rho^{{-1}}(\overline{{B}}_1,...,\overline{{B}}_d)\in L^2(G;{{\mathbb R}}^d,m)}}$\end{document} satisfies \documentclass[12pt]{minimal}
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\begin{document}$${{ \sum_{{i=1}}^{{d}}\int_G \overline{{B}}_i \partial_i f\,dx =0\quad {{\rm{ for all}}} f\in C^{{\infty}}(\overline{{G}}), }}$$\end{document} are constructed and analyzed. If G is a bounded Lipschitz domain, ρ∈H1,1(G), and aij∈D(ℰr), a Skorokhod decomposition for Xt is given. This happens through a local time that is uniquely associated to the smooth measure 1{Tr(ρ)>0}dΣ, where Tr denotes the trace and Σ the surface measure on ∂G.
