This is a thorough and accessible exposition on the functional analytic approach to the problem of construction of Markov processes with Ventcel' boundary conditions in probability theory. It presents new developments in the theory of singular integrals.
This volume is devoted to a thorough and accessible exposition on the functional analytic approach to the problem of construction of Markov processes with Ventcel' boundary conditions in probability theory. Analytically, a Markovian particle in a domain of Euclidean space is governed by an integro-differential operator, called a Waldenfels operator, in the interior of the domain, and it obeys a boundary condition, called the Ventcel' boundary condition, on the boundary of the domain. Probabilistically, a Markovian particle moves both by jumps and continuously in the state space and it obeys the Ventcel' boundary condition, which consists of six terms corresponding to the diffusion along the boundary, the absorption phenomenon, the reflection phenomenon, the sticking (or viscosity) phenomenon, the jump phenomenon on the boundary, and the inward jump phenomenon from the boundary. In particular, second-order elliptic differential operators are called diffusion operators and describe analytically strong Markov processes with continuous paths in the state space such as Brownian motion. We observe that second-order elliptic differential operators with smooth coefficients arise naturally in connection with the problem of construction of Markov processes in probability. Since second-order elliptic differential operators are pseudo-differential operators, we can make use of the theory of pseudo-differential operators as in the previous book: Semigroups, boundary value problems and Markov processes (Springer-Verlag, 2004).
Our approach here is distinguished by its extensive use of the ideas and techniques characteristic of the recent developments in the theory of partial differential equations. Several recent developments in the theory of singular integrals have made further progress in the study of elliptic boundary value problems and hence in the study of Markov processes possible. The presentation of these new results is the main purpose of this book.
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
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.
Inhaltsverzeichnis
and Main Results.- Semigroup Theory.- L Theory of Pseudo-Differential Operators.- L Approach to Elliptic Boundary Value Problems.- Proof of Theorem 1.1.- A Priori Estimates.- Proof of Theorem 1.2.- Proof of Theorem 1.3 - Part (i).- Proof of Theorem 1.3, Part (ii).- Application to Semilinear Initial-Boundary Value Problems.- Concluding Remarks.
Klappentext
ti?cResearch(No.19540162),MinistryofEducation,Culture,Sports,Science and Technology, Japan.
Includes supplementary material: sn.pub/extras
