Reflecting Diffusions on Lipschitz Domains with Cusps -Analytic Construction and Skorohod Representation-

Abstract

The reflecting Brownian motion on a bounded Lipschitz domain D ⊂ R d was constructed by Bass and Hsu [3] as a conservative, symmetric (with respect to the Lebesgue measure) and strong Feller diffusion process (X t ,P x ) on the closure \( \overline D \) of D whose transition semigroup on L 2 (D) is associated with the Sobolev space H 1 (D) with inner product $$ \varepsilon \left( {u,v} \right) = \frac{1}{2}\int_D {\nabla u \cdot \nabla vdx,} \quad \quad u,v \in {H^1}\left( D \right)$$ (1.1) .