Feller Semigroups and Boundary Value Problems

TheBoundary value problem purposeFeller semigroup of this chapter is to prove a general existence theorem for Feller semigroups in terms of boundary value problems (Theorem 24.9), following the main idea of Taira [148, Sect. 9.6] (cf. Bony–Courrège–Priouret [15] and Sato–Ueno [124]). Intuitively, Theorem 24.9 asserts that we can “piece together” a Markov process on the boundary \(\partial \Omega \) with A-diffusion in the interior \(\Omega \) to construct a Markov process on the closure \(\overline{\Omega } = \Omega \cup {\partial \Omega }\) (see Remark 24.5).