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

摘要

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, ${{gin C^{{infty}}(overline{{G}})}}$ , is closable in L 2 (G,m) with closure (ℰ r ,D(ℰ r )). 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 ${{ {{{{mathcal E}}}}^r(f,g) -sum_{{i=1}}^{{d}}int_G rho^{{-1}}overline{{B}}_ipartial_i f, g, dm; f,gin D({{{{mathcal E}}}}^r)_b }}$ where ${{rho^{{-1}}(overline{{B}}_1,...,overline{{B}}_d)in L^2(G;{{mathbb R}}^d,m)}}$ satisfies ${{ sum_{{i=1}}^{{d}}int_G overline{{B}}_i partial_i f,dx =0quad {{rm{ for all}}} fin C^{{infty}}(overline{{G}}), }}$ are constructed and analyzed. If G is a bounded Lipschitz domain, ρ∈H 1,1 (G), and a ij ∈D(ℰ 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 (ρ)>0} dΣ, where Tr denotes the trace and Σ the surface measure on ∂G.