Singular integrals and Feller semigroups with jump phenomena

This paper is devoted to the real analysis methods for the problem of construction of Markov processes with boundary conditions in probability. Analytically, a Markovian particle in a domain of Euclidean space is governed by an integro-differential operator, called Waldenfels operator, in the interior of the domain, and it obeys a boundary condition, called Ventcel' (Wentzell) boundary condition, on the boundary of the domain. Probabilistically, a Markovian particle moves both by continuous paths and by jumps in the state space and it obeys the Ventcel' boundary condition that consists of six terms corresponding to a diffusion along the boundary, an absorption phenomenon, a reflection phenomenon, a sticking (or viscosity) phenomenon and a jump phenomenon on the boundary and an inward jump phenomenon from the boundary. More precisely, we study a class of first order Ventcel' boundary value problems for second order elliptic Waldenfels integro-differential operators. By using the Calderon-Zygmund theory of singular integrals in real analysis, we prove existence and uniqueness theorems in the framework of Sobolev and Besov spaces, which extend earlier theorems due to Bony-Courrege-Priouret to the VMO (vanishing mean oscillation) case. Our proof is based on various maximum principles for second order elliptic Waldenfels operators with discontinuous coefficients in the framework of Sobolev spaces.