<p>This paper is devoted to a careful and accessible exposition of the real and functional analytic methods for the problem of construction of strong Markov processes on a bounded domain in Euclidean space with boundary. Our approach is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of A.&#xa0;P. Calderón and A. Zygmund of singular integral operators with non-smooth kernels. We construct Feller semigroups with Dirichlet condition for elliptic Waldenfels integro-differential operators with discontinuous coefficients. Intuitively, we construct a Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle moves both by continuous paths and by jumps in the state space until it dies at the time when it reaches the boundary. This paper will lead to a deep insight into the study of elliptic boundary value problems for elliptic Waldenfels integro-differential operators with discontinuous coefficients in the theory of partial differential equations. The author declares no conflict of interest to disclose.</p>