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.
Keywords
- Dirichlet Problem
- Uniqueness Theorem
- Bounded Linear Operator
- Dense Subset
- Resolvent Equation
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© 2009 Springer-Verlag Berlin Heidelberg
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Taira, K. (2009). Proof of Theorem 1.3, Part (ii). In: Boundary Value Problems and Markov Processes. Lecture Notes in Mathematics(), vol 1499. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01677-6_9
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DOI: https://doi.org/10.1007/978-3-642-01677-6_9
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-01676-9
Online ISBN: 978-3-642-01677-6
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