Proofs of Theorem 1.5 and Theorem 1.4, Part (ii)

Abstract

In this chapter we prove Theorem 1.5 and part (ii) of Theorem 1.4. First, we study Feller semigroups with reflecting barrier (Theorem 13.14) and then, using these Feller semigroups we construct Feller semigroups corresponding to such a physical phenomenon that either absorption or reflection phenomenon occurs at each point of the boundary (Theorem 13.18). Our proof is based on the generation theorems for Feller semigroups discussed in Sect. 3.3, just as in Chap. 7. Part (i) of Theorem 1.3, together with Theorem 1.5, proves part (ii) of Theorem 1.4.