Semigroups, Boundary Value Problems and Markov Processes

ISBN-13: 9783642073717 / Angielski / Miękka / 2010 / 340 str.

Kazuaki Taira

Opis

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.

Recenzje

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)

Spis treści

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

O autorze

 

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.