Skorokhod decomposition of reflected diffusions on bounded Lipschitz domains with singular non-reflection part

Let G(subset of)R^d be a compact set with interior G. Let (rho)(element of)L ^1 (G,dx), (rho)>0 dx-a.e. on G, and m:=(rho)dx. Let A=(a ij ) be symmetric, and globally uniformly strictly elliptic on G. Let (rho)be such that (sigma)^r(f,g)=1/2(sigma)(integral)G a ij ((partial defferential)i g f (partial defferential)j g d m; f, g(element of) C^(infinity)(G), is closable in L 2 (G,m) with closure ((sigma)^ r ,D((sigma)^ r )). The latter is fulfilled if (rho) satisfies the Hamza type condition, or (partial defferenial)i (rho)(element of)L 1 loc (G,dx), 1<=i<=d. Conservative, non-symmetric diffusion processes X t related to the extension of a generalized Dirichlet form (sigma)^r (f,g)(sigma)(integral)(rho)^-1 B I (partial defferential) i f g dm; f,g(element of)D((sigma)^r b where (rho )^-1 (B1,…,B d)(element of) L^2(G;R^d,m)satisfies (sigma)(integral)B i (partial defferential)i f d x =0 for all f(element of) C^(infinity)(G,are constructed and analyzed. If G is a bounded Lipschitz domain, (rho) (element of) H ^1,1 (G), and a ij (element of)D((sigma)^ r ), a Skorokhod decomposition for X t is given. This happens through a local time that is uniquely associated to the smooth measure 1{ Tr ((rho))>0} d(sigma), where Tr denotes the trace and (sigma) the surface measure on (partial defferential) G.