Semigroups, Boundary Value Problems and Markov Processes

This volume will be of great appeal to both advanced students and researchers. For the former, it serves as an effective introduction to three interrelated subjects of analysis: semigroups, Markov processes and elliptic boundary value problems. For the latter, it provides a new method for the analysis of Markov processes, a powerful method clearly capable of extensive further development.

Table of Contents:
Preface Introduction and Main Results Chapter 1 Theory of Semigroups Section 1.1 Banach Space Valued Functions Section 1.2 Operator Valued Functions Section 1.3 Exponential Functions Section 1.4 Contraction Semigroups Section 1.5 Analytic Semigroups Chapter 2 Markov Processes and Semigroups Section 2.1 Markov Processes Section 2.2 Transition Functions and Feller Semigroups Section 2.3 Generation Theorems for Feller Semigroups Section 2.4 Borel Kernels and the Maximum Principle Chapter 3 Theory of Distributions Section 3.1 Notation Section 3.2 L^p Spaces Section 3.3 Distributions Section 3.4 The Fourier Transform Section 3.5 Operators and Kernels Section 3.6 Layer Potentials Subsection 3.6.1 The Jump Formula Subsection 3.6.2 Single and Double Layer Potentials Subsection 3.6.3 The Green Representation Formula Chapter 4 Theory of Pseudo-Differential Operators Section 4.1 Function Spaces Section 4.2 Fourier Integral Operators Subsection 4.2.1 Symbol Classes Subsection 4.2.2 Phase Functions Subsection 4.2.3 Oscillatory Integrals Subsection 4.2.4 Fourier Integral Operators Section 4.3 Pseudo-Differential Operators Section 4.4 Potentials and Pseudo-Differential Operators Section 4.5 The Transmission Property Section 4.6 The Boutet de Monvel Calculus Appendix A Boundedness of Pseudo-Differential Operators Section A.1 The Littlewood--Paley Series Section A.2 Definition of Sobolev and Besov Spaces Section A.3 Non-Regular Symbols Section A.4 The L^p Boundedness Theorem Section A.5 Proof of Proposition A.1 Section A.6 Proof of Proposition A.2 Chapter 5 Elliptic Boundary Value Problems Section 5.1 The Dirichlet Problem Section 5.2 Formulation of a Boundary Value Problem Section 5.3 Reduction to the Boundary Chapter 6 Elliptic Boundary Value Problems and Feller Semigroups Section 6.1 Formulation of a Problem Section 6.2 Transversal Case Subsection 6.2.1 Generation Theorem for Feller Semigroups Subsection 6.2.2 Sketch of Proof of Theorem 6.1 Subsection 6.2.3 Proof of Theorem 6.15 Section 6.3 Non-Transversal Case Subsection 6.3.1 The Space C_0( \ M) Subsection 6.3.2 Generation Theorem for Feller Semigroups Subsection 6.3.3 Sketch of Proof of Theorem 6.20 Appendix B Unique Solvability of Pseudo-Differential Operators Chapter 7 Proof of Theorem 1 Section 7.1 Regularity Theorem for Problem (0.1) Section 7.2 Uniqueness Theorem for Problem (0.1) Section 7.3 Existence Theorem for Problem (0.1) Subsection 7.3.1 Proof of Theorem 7.7 Subsection 7.3.2 Proof of Proposition 7.10 Chapter 8 Proof of Theorem 2 Chapter 9 A Priori Estimates Chapter 10 Proof of Theorem 3 Section 10.1 Proof of Part (i) of Theorem 3 Section 10.2 Proof of Part (ii) of Theorem 3 Chapter 11 Proof of Theorem 4, Part (i) Section 11.1 Sobolev's Imbedding Theorems Section 11.2 Proof of Part (i) of Theorem 4 Chapter 12 Proofs of Theorem 5 and Theorem 4, Part (ii) Section 12.1 Existence Theorem for Feller Semigroups Section 12.2 Feller Semigroups with Reflecting Barrier Section 12.3 Proof of Theorem 5 Section 12.4 Proof of Part (ii) of Theorem 4 Chapter 13 Boundary Value Problems for Waldenfels Operators Section 13.1 Formulation of a Boundary Value Problem Section 13.2 Proof of Theorem 6 Section 13.3 Proof of Theorem 7 Section 13.4 Proof of Theorem 8 Section 13.5 Proof of Theorem 9 Section 13.6 Concluding Remarks