Stochastic differential equations with sticky reflection and boundary diffusion

Abstract

We construct diffusion processes in bounded domains $\Omega $ with sticky reflection at the boundary $\Gamma $ in use of Dirichlet forms. In particular, the occupation time on the boundary is positive. The construction covers a static boundary behavior and an optional diffusion along $\Gamma $. The process is a solution to a given SDE for q.e. starting point. Using regularity results for elliptic PDE with Wentzell boundary conditions we show strong Feller properties and characterize the constructed process even for every starting point in $\overline{\Omega } \backslash \Xi $, where $\Xi $ is given explicitly by the involved densities. By a time change we obtain pointwise solutions to SDEs with immediate reflection under weak assumptions on $\Gamma $ and the drift. A non-trivial extension of the construction yields N-particle systems with the stated boundary behavior and singular drifts. Finally, the setting is applied to a model for particles diffusing in a chromatography tube with repulsive interactions.