Semigroups, boundary value problems and Markov processes /

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Bibliographic Details
Author / Creator:Taira, Kazuaki.
Imprint:Berlin ; New York : Springer, c2004.
Description:xi, 337 p. : ill. ; 24 cm.
Language:English
Series:Springer monographs in mathematics, 1439-7382
Subject:Markov processes.
Semigroups.
Boundary value problems.
Boundary value problems.
Markov processes.
Semigroups.
Format: Print Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/5055873
Hidden Bibliographic Details
ISBN:3540406514 (acid-free paper)
Notes:Includes bibliographical references (p. [325]-328) and indexes.
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
  • Appendix C. The Maximum Principle
  • Section C.1. The Weak Maximum Principle
  • Section C.2. The Strong Maximum Principle
  • Section C.3. The Boundary Point Lemma
  • Bibliography
  • Index of Symbols
  • Subject Index