Abstract.
Let \({{\overline{{G}}\subset {{\mathbb R}}^d}}\) be a compact set with interior G. Let ρ∈L 1(G,dx), ρ>0 dx-a.e. on G, and m:=ρdx. Let A=(a ij ) be symmetric, and globally uniformly strictly elliptic on G. Let ρ be such that \({{{{{{\mathcal E}}}}^r(f,g)=\frac{{1}}{{2}}\sum_{{i,j=1}}^{{d}}\int_G a_{{ij}}\partial_i f \partial_j g\,dm}}\); f, \({{g\in C^{{\infty}}(\overline{{G}})}}\), is closable in L 2(G,m) with closure (ℰr,D(ℰr)). The latter is fulfilled if ρ satisfies the Hamza type condition, or ∂ i ρ∈L 1 loc (G,dx), 1≤i≤d. Conservative, non-symmetric diffusion processes X t related to the extension of a generalized Dirichlet form \({{ {{{{\mathcal E}}}}^r(f,g) -\sum_{{i=1}}^{{d}}\int_G \rho^{{-1}}\overline{{B}}_i\partial_i f\, g\, dm; f,g\in D({{{{\mathcal E}}}}^r)_b }}\) where \({{\rho^{{-1}}(\overline{{B}}_1,...,\overline{{B}}_d)\in L^2(G;{{\mathbb R}}^d,m)}}\) satisfies \({{ \sum_{{i=1}}^{{d}}\int_G \overline{{B}}_i \partial_i f\,dx =0\quad {{\rm{ for all}}} f\in C^{{\infty}}(\overline{{G}}), }}\) are constructed and analyzed. If G is a bounded Lipschitz domain, ρ∈H 1,1(G), and a ij ∈D(ℰr), a Skorokhod decomposition for X t 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.
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This research has been financially supported by TMR grant HPMF-CT-2000-00942 of the European Union.
Mathematics Subject Classification (2000): 60J60, 60J55, 31C15, 31C25, 35J25
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Trutnau, G. Skorokhod decomposition of reflected diffusions on bounded Lipschitz domains with singular non-reflection part. Probab. Theory Relat. Fields 127, 455–495 (2003). https://doi.org/10.1007/s00440-003-0296-9
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DOI: https://doi.org/10.1007/s00440-003-0296-9