Proof of Theorem 1.4, Part (i)

Abstract

In this chapter we prove Theorem 1.4 and part (ii) of Theorem 1.3. This chapter is the heart of the subject. General existence theorems for Feller semigroups are formulated in terms of elliptic boundary value problems with spectral parameter (Theorem 9.12). First, we study Feller semigroups with reflecting barrier (Theorem 9.14) and then, by using these Feller semigroups we construct Feller semigroups corresponding to such a diffusion phenomenon that either absorption or reflection phenomenon occurs at each point of the boundary (Theorem 9.18). Our proof is based on the generation theorems of Feller semigroups discussed in Section 2.2.