Oblique derivative problems and Feller semigroups with discontinuous coefficients

Abstract

This paper is devoted to the functional analytic approach to the problem of existence of Markov processes with an oblique derivative boundary condition for second-order, uniformly elliptic differential operators with discontinuous coefficients. More precisely, we construct Feller semigroups associated with absorption, reflection, drift and sticking (or viscosity) phenomena at the boundary. The approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the Calderon–Zygmund theory of singular integral operators with non-smooth kernels.