On reflecting diffusion processes and Skorokhod decompositions

Abstract

Let G be a d-dimensional bounded Euclidean domain, H1 (G) the set of f in L2(G) such that ∇f (defined in the distribution sense) is in L2(G). Reflecting diffusion processes associated with the Dirichlet spaces (H1(G), ℰ) on L2(G, σd x) are considered in this paper, where[Figure not available: see fulltext.] A=(aij is a symmetric, bounded, uniformly elliptic d×d matrix-valued function such that aij∈H1(G) for each i,j, and σ∈H1(G) is a positive bounded function on G which is bounded away from zero. A Skorokhod decomposition is derived for the continuous reflecting Markov processes associated with (H1(G), ℰ) having starting points in G under a mild condition which is satisfied when π{variant}G has finite (d-1)-dimensional lower Minkowski content. © 1993 Springer-Verlag.