Proofs of Theorem 1.5, Part (ii) and Theorem 1.6

Abstract

In this chapter we prove Theorem 1.6 in Section 13.3 and part (ii) of Theorem 1.5 in Section 13.4. This chapter is the heart of the subject. In Section 13.1 general existence theorems for Feller semigroups are formulated in terms of elliptic boundary value problems with spectral parameter (Theorem 13.4). In Section 13.1 we study Feller semigroups with reflecting barrier (Theorem 13.17) 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 13.22). Our proof is based on the generation theorems of Feller semigroups discussed in Chapter 3.