On the existence of Feller semigroups with discontinuous coefficients (Q2508576)

The main result of this paper is the construction of Feller semigroups with Dirichlet conditions for second-order uniformly elliptic integro-differential operators \(A+W\) with VMO (thus, possibly discontinuous!) coefficients. The approach is functional-analytic and uses an appropriate version of the Hille--Yosida(--Ray) theorem. The central point is to solve a Dirichlet problem in an \(L^p\)-context for the diffusion (i.e., differential-operator) part \(A\), resp.\ the full integro-differential operator \(A+W\) (consisting of the diffusion operator \(A\) plus a von Waldenfels-type \(W\) integro-differential operator). The point of view of this paper is to understand (in terms of \(L^p\)-Sobolev spaces) \(W\) as a small perturbation of \(A\).