Dirichlet problems with discontinuous coefficients and Feller semigroups

Abstract

This paper is devoted to the functional analytic approach to the problem of existence of Markov processes in probability theory. More precisely, we construct Feller semigroups with Dirichlet conditions for second-order, uniformly elliptic integro-differential operators with discontinuous coefficients. Intuitively, we prove that there exists a Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle moves both by jumps and continuously in the state space until it dies at the time when it reaches the boundary. Our approach is based on real analysis techniques.