On the existence of Feller semigroups with discontinuous coefficients II

Abstract

This paper is devoted to the functional analytic approach to the problem of existence of Markov processes with Dirichlet boundary condition, oblique derivative boundary condition and first-order Wentzell 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 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 Calderón-Zygmund theory of singular integral operators with non-smooth kernels.