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

• Published in: Probability theory and related fields Vol. 127; no. 4; pp. 455 - 495
• Main Author: TRUTNAU, Gerald
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer 01.12.2003; Berlin Springer Nature B.V; New York, NY
• Subjects: Decomposition; Diffusion; Digital Object Identifier; Dirichlet problem; Domains; Exact sciences and technology; Markov analysis; Markov processes; Mathematical analysis; Mathematics; Partial differential equations; Potential theory; Probability; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Second order; Lipschitz domain; Skorokhod decomposition; Dirichlet space; Decomposition; Potential; Stochastic process; Local time; Additive functional; Boundary value problem; Bounded domain; Capacity; Dirichlet form; Diffusion process; Elliptic operator; Diffusion
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-003-0296-9

Let 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 ; f, , 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 where satisfies 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.