Synopses & Reviews
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.
Review
From the reviews: "This book is devoted to the study of certain uniformly elliptic boundary value problems and associated semigroups. ... The main results are all laid out in the introduction, so it is always clear where the book is headed. ... For the probabilist, the book provides a good introduction to modern sophisticated results on analytical problems associated with diffusion processes with possibly additional Lèvy-type jumps. In some cases, the book may also be a useful reference ... ." (Jan M. Swart, Jahresberichte der Deutschen Mathematiker Vereinigung, November, 2005) "This book by Kazuaki Taira contains a detailed study of semigroups, elliptical boundary value problems, Markov processes and the relations between these mathematical concepts. ... The book grew out of a series of lectures and lecture notes; this facilitates its use for teaching at the graduate level. The presentation is detailed and clear ... . I would recommend the book for graduate students or researchers interested mainly in the analytical aspects of Markov process theory ... ." (R. Frey, ZAA - Zeitschrift für Analysis und ihre Anwendungen, Vol. 23 (3), 2004) "In this book the author proposes the study of three interrelated subjects in analysis: semigroups, elliptic boundary value problems and Markov processes. ... The well chosen material given in an appropriate form and style makes the book very useful for the university students as well as for mathematicians with interests in probability theory, functional analysis and partial differential equations." (Mikhail P. Moklyachuk, Zentralblatt MATH, Vol. 1035, 2004) "The book is devoted to the generation of analytic Feller semigroups by operators corresponding to boundary value problems for second order elliptic differential and integro-differential equations. ... the present book is a valuable contribution to a rich field of mathematics emerging at the interface of functional analysis, partial differential equations, and stochastic processes." (Anatoly N. Kochubei, Mathematical Reviews, 2004 i)
Review
From the reviews:
"This book is devoted to the study of certain uniformly elliptic boundary value problems and associated semigroups. ... The main results are all laid out in the introduction, so it is always clear where the book is headed. ... For the probabilist, the book provides a good introduction to modern sophisticated results on analytical problems associated with diffusion processes with possibly additional Lèvy-type jumps. In some cases, the book may also be a useful reference ... ." (Jan M. Swart, Jahresberichte der Deutschen Mathematiker Vereinigung, November, 2005)
"This book by Kazuaki Taira contains a detailed study of semigroups, elliptical boundary value problems, Markov processes and the relations between these mathematical concepts. ... The book grew out of a series of lectures and lecture notes; this facilitates its use for teaching at the graduate level. The presentation is detailed and clear ... . I would recommend the book for graduate students or researchers interested mainly in the analytical aspects of Markov process theory ... ." (R. Frey, ZAA - Zeitschrift für Analysis und ihre Anwendungen, Vol. 23 (3), 2004)
"In this book the author proposes the study of three interrelated subjects in analysis: semigroups, elliptic boundary value problems and Markov processes. ... The well chosen material given in an appropriate form and style makes the book very useful for the university students as well as for mathematicians with interests in probability theory, functional analysis and partial differential equations." (Mikhail P. Moklyachuk, Zentralblatt MATH, Vol. 1035, 2004)
"The book is devoted to the generation of analytic Feller semigroups by operators corresponding to boundary value problems for second order elliptic differential and integro-differential equations. ... the present book is a valuable contribution to a rich field of mathematics emerging at the interface of functional analysis, partial differential equations, and stochastic processes." (Anatoly N. Kochubei, Mathematical Reviews, 2004 i)
Synopsis
The purpose of this book is to provide a careful and accessible account along modern lines of the subject wh ich the title deals, as weIl as to discuss prob lems of current interest in the field. Unlike many other books on Markov processes, this book focuses on the relationship between Markov processes and elliptic boundary value problems, with emphasis on the study of analytic semigroups. More precisely, this book is devoted to the functional analytic approach to a class of degenerate boundary value problems for second-order elliptic integro-differential operators, called Waldenfels operators, whi: h in cludes as particular cases the Dirichlet and Robin problems. We prove that this class of boundary value problems provides a new example of analytic semi groups both in the LP topology and in the topology of uniform convergence. As an application, we construct a strong Markov process corresponding to such a physical phenomenon that a Markovian particle moves both by jumps and continuously in the state space until it "dies" at the time when it reaches the set where the particle is definitely absorbed. The approach here is distinguished by the extensive use of the techniques characteristic of recent developments in the theory of partial differential equa tions. The main technique used is the calculus of pseudo-differential operators which may be considered as a modern theory of potentials."
About the Author
Kazuaki TAIRA is Professor of Mathematics at the University of Tsukuba, Japan, where he has taught since 1998. He received his Bachelor of Science (1969) degree from the University of Tokyo, Japan, and his Master of Science (1972) degree from Tokyo Institute of Technology, Japan, where he served as an Assistant between 1972-1978. He holds the Doctor of Science (1976) degree from the University of Tokyo, and the Doctorat d'Etat (1978) degree from Université de Paris-Sud, France, where he received a French Government Scholarship in 1976-1978. Dr. Taira was also a member of the Institute for Advanced Study, U. S. A., in 1980-1981. He was Associate Professor of the University of Tsukuba between 1981-1995, and Professor of Hiroshima University, Japan, between 1995-1998.
His current research interests are in the study of three interrelated subjects in analysis: semigroups, elliptic boundary value problems and Markov processes.
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 5.3 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