This book is an easy-to-read reference providing a link between functional analysis and diffusion processes. More precisely, the book takes readers to a mathematical crossroads of functional analysis (macroscopic approach), partial differential equations (mesoscopic approach), and probability (microscopic approach) via the mathematics needed for the hard parts of diffusion processes. This work brings these three fields of analysis together and provides a profound stochastic insight (microscopic approach) into the study of elliptic boundary value problems.
The author does a massive study of diffusion processes from a broad perspective and explains mathematical matters in a more easily readable way than one usually would find. The book is amply illustrated; 14 tables and 141 figures are provided with appropriate captions in such a fashion that readers can easily understand powerful techniques of functional analysis for the study of diffusion processes in probability.
The scope of the author’s work has been and continues to be powerful methods of functional analysis for future research of elliptic boundary value problems and Markov processes via semigroups. A broad spectrum of readers can appreciate easily and effectively the stochastic intuition that this book conveys. Furthermore, the book will serve as a sound basis both for researchers and for graduate students in pure and applied mathematics who are interested in a modern version of the classical potential theory and Markov processes.
For advanced undergraduates working in functional analysis, partial differential equations, and probability, it provides an effective opening to these three interrelated fields of analysis. Beginning graduate students and mathematicians in the field looking for a coherent overview will find the book to be a helpful beginning.
This work will be a major influence in a very broad field of study for a long time.
Author(s): Kazuaki Taira
Series: Springer Monographs in Mathematics
Edition: 1
Publisher: Springer Nature Singapore
Year: 2022
Language: English
Pages: 782
City: Singapore
Tags: Diffusion process, Markov process, Feller semigroup, Elliptic boundary value problem, Pseudo-differential operator
Foreword
Preface
Contents
Notation and Conventions
1 Introduction and Summary
1.1 Markov Processes and Semigroups
1.1.1 Brownian Motion
1.1.2 Markov Processes
1.1.3 Transition Functions
1.1.4 Kolmogorov's Equations
1.1.5 Feller Semigroups
1.1.6 Path Functions of Markov Processes
1.1.7 Strong Markov Processes
1.1.8 Infinitesimal Generators of Feller Semigroups
1.1.9 One-Dimensional Diffusion Processes
1.1.10 Multidimensional Diffusion Processes
1.2 Propagation of Maxima
1.3 Construction of Feller Semigroups
1.4 Notes and Comments
Part I Foundations of Modern Analysis
2 Sets, Topology and Measures
2.1 Sets
2.2 Mappings
2.3 Topological Spaces
2.4 Compactness
2.5 Connectedness
2.6 Metric Spaces
2.7 Baire's Category
2.8 Continuous Mappings
2.9 Linear Spaces
2.10 Linear Topological Spaces
2.10.1 The Ascoli–Arzelà Theorem
2.11 Factor Spaces
2.12 Algebras and Modules
2.13 Linear Operators
2.14 Measurable Spaces
2.15 Measurable Functions
2.16 Measures
2.16.1 Lebesgue Measures
2.16.2 Signed Measures
2.16.3 Borel Measures and Radon Measures
2.16.4 Product Measures
2.16.5 Direct Image of Measures
2.17 Integrals
2.18 The Radon–Nikodým Theorem
2.19 Fubini's Theorem
2.20 Notes and Comments
3 A Short Course in Probability Theory
3.1 Measurable Spaces and Functions
3.1.1 The Monotone Class Theorem
3.1.2 The Approximation Theorem
3.1.3 Measurability of Functions
3.2 Probability Spaces
3.3 Random Variables and Expectations
3.4 Independence
3.4.1 Independent Events
3.4.2 Independent Random Variables
3.4.3 Independent Algebras
3.5 Construction of Random Processes with Finite Dimensional Distribution
3.6 Conditional Probabilities
3.7 Conditional Expectations
3.8 Notes and Comments
4 Manifolds, Tensors and Densities
4.1 Manifolds
4.1.1 Topology on Manifolds
4.1.2 Submanifolds
4.2 Smooth Mappings
4.2.1 Partitions of Unity
4.3 Tangent Bundles
4.4 Vector Fields
4.5 Vector Fields and Integral Curves
4.6 Cotangent Bundles
4.7 Tensors
4.8 Tensor Fields
4.9 Exterior Product
4.10 Differential Forms
4.11 Vector Bundles
4.12 Densities
4.13 Integration on Manifolds
4.14 Manifolds with Boundary and the Double of a Manifold
4.15 Stokes's Theorem, Divergence Theorem and Green's Identities
4.16 Notes and Comments
5 A Short Course in Functional Analysis
5.1 Metric Spaces and the Contraction Mapping Principle
5.2 Linear Operators and Functionals
5.3 Quasinormed Linear Spaces
5.3.1 Compact Sets
5.3.2 Bounded Sets
5.3.3 Continuity of Linear Operators
5.3.4 Topologies of Linear Operators
5.3.5 The Banach–Steinhaus Theorem
5.3.6 Product Spaces
5.4 Normed Linear Spaces
5.4.1 Linear Operators on Normed Spaces
5.4.2 Method of Continuity
5.4.3 Finite Dimensional Spaces
5.4.4 The Hahn–Banach Extension Theorem
5.4.5 Dual Spaces
5.4.6 Annihilators
5.4.7 Dual Spaces of Normed Factor Spaces
5.4.8 Bidual Spaces
5.4.9 Weak Convergence
5.4.10 Weak* Convergence
5.4.11 Dual Operators
5.4.12 Adjoint Operators
5.5 Linear Functionals and Measures
5.5.1 The Space of Continuous Functions
5.5.2 The Space of Signed Measures
5.5.3 The Riesz–Markov Representation Theorem
5.5.4 Weak Convergence of Measures
5.6 Closed Operators
5.7 Complemented Subspaces
5.8 Compact Operators
5.9 The Riesz–Schauder Theory
5.10 Fredholm Operators
5.11 Hilbert Spaces
5.11.1 Orthogonality
5.11.2 The Closest-Point Theorem and Applications
5.11.3 Orthonormal Sets
5.11.4 Adjoint Operators
5.12 The Hilbert–Schmidt Theory
5.13 Notes and Comments
6 A Short Course in Semigroup Theory
6.1 Banach Space Valued Functions
6.2 Operator Valued Functions
6.3 Exponential Functions
6.4 Contraction Semigroups
6.4.1 The Hille–Yosida Theory of Contraction Semigroups
6.4.2 The Contraction Semigroup Associated with the Heat Kernel
6.5 (C0) Semigroups
6.5.1 Semigroups and Their Infinitesimal Generators
6.5.2 Infinitesimal Generators and Their Resolvents
6.5.3 The Hille–Yosida Theorem
6.5.4 (C0) Semigroups and Initial-Value Problems
6.6 Notes and Comments
Part II Elements of Partial Differential Equations
7 Distributions, Operators and Kernels
7.1 Notation
7.1.1 Points in Euclidean Spaces
7.1.2 Multi-Indices and Derivations
7.2 Function Spaces
7.2.1 Lp Spaces
7.2.2 Convolutions
7.2.3 Spaces of Ck Functions
7.2.4 Space of Test Functions
7.2.5 Hölder Spaces
7.2.6 Friedrichs' Mollifiers
7.3 Differential Operators
7.4 Distributions and the Fourier Transform
7.4.1 Definitions and Basic Properties of Distributions
7.4.2 Topologies on mathcalD(Ω)
7.4.3 Support of a Distribution
7.4.4 Dual Space of Cinfty(Ω)
7.4.5 Tensor Product of Distributions
7.4.6 Convolution of Distributions
7.4.7 The Jump Formula
7.4.8 Regular Distributions with Respect to One Variable
7.4.9 The Fourier Transform
7.4.10 Tempered Distributions
7.4.11 Fourier Transform of Tempered Distributions
7.5 Operators and Kernels
7.5.1 Schwartz's Kernel Theorem
7.5.2 Regularizers
7.6 Layer Potentials
7.6.1 Single and Double Layer Potentials
7.6.2 The Green Representation Formula
7.6.3 Approximation to the Identity via Dirac Measure
7.7 Distribution Theory on a Manifold
7.7.1 Densities on a Manifold
7.7.2 Distributions on a Manifold
7.7.3 Differential Operators on a Manifold
7.7.4 Operators and Kernels on a Manifold
7.8 Domains of Class Cr
7.9 The Seeley Extension Theorem
7.9.1 Proof of Lemma 7.46
7.10 Notes and Comments
8 L2 Theory of Sobolev Spaces
8.1 The Spaces Hs(Rn)
8.2 The Spaces Hsloc(Ω) and Hscomp(Ω)
8.3 The Spaces Hs(M)
8.4 The Spaces Hs(overlineRn+)
8.5 The Spaces Hs(overlineΩ)
8.6 Trace Theorems
8.7 Sectional Trace Theorems
8.8 Sobolev Spaces and Regularizations
8.9 Friedrichs' Mollifiers and Differential Operators
8.10 Notes and Comments
9 L2 Theory of Pseudo-differential Operators
9.1 Symbol Classes
9.2 Phase Functions
9.3 Oscillatory Integrals
9.4 Fourier Integral Operators
9.5 Pseudo-differential Operators
9.5.1 Definitions and Basic Properties
9.5.2 Symbols of a Pseudo-differential Operator
9.5.3 The Algebra of Pseudo-differential Operators
9.5.4 Elliptic Pseudo-differential Operators
9.5.5 Invariance of Pseudo-differential Operators Under Change of Coordinates
9.5.6 Pseudo-differential Operators and Sobolev Spaces
9.6 Pseudo-differential Operators on a Manifold
9.6.1 Definitions and Basic Properties
9.6.2 Classical Pseudo-differential Operators
9.6.3 Elliptic Pseudo-differential Operators
9.7 Elliptic Pseudo-differential Operators and Their Indices
9.7.1 Pseudo-differential Operators on Sobolev Spaces
9.7.2 The Index of an Elliptic Pseudo-differential Operator
9.8 Potentials and Pseudo-differential Operators
9.8.1 Single and Double Layer Potentials Revisited
9.8.2 The Green Representation Formula Revisited
9.8.3 Surface and Volume Potentials
9.9 The Sharp Gårding Inequality
9.10 Hypoelliptic Pseudo-differential Operators
9.11 Notes and Comments
Part III Maximum Principles and Elliptic Boundary Value Problems
10 Maximum Principles for Degenerate Elliptic Operators
10.1 Introduction
10.2 Maximum Principles
10.3 Propagation of Maxima
10.3.1 Statement of Results
10.3.2 Preliminaries
10.3.3 Proof of Theorem 10.14
10.3.4 Proof of Theorem 10.19
10.3.5 Proof of Theorem 10.17
10.4 Notes and Comments
Part IV L2 Theory of Elliptic Boundary Value Problems
11 Elliptic Boundary Value Problems
11.1 The Dirichlet Problem in the Framework of Hölder Spaces
11.2 The Dirichlet Problem in the Framework of L2 Sobolev Spaces
11.3 General Boundary Value Problems
11.3.1 Formulation of Boundary Value Problems
11.3.2 Reduction to the Boundary
11.4 Unique Solvability Theorem for General Boundary Value Problems
11.4.1 Statement of Main Results
11.4.2 Proof of Theorem11.19
11.4.3 End of Proof of Theorem11.19
11.4.4 Proof of Corollary11.20
11.5 Notes and Comments
Part V Markov Processes, Feller Semigroups and Boundary Value Problems
12 Markov Processes, Transition Functions and Feller Semigroups
12.1 Markov Processes and Transition Functions
12.1.1 Definitions of Markov Processes
12.1.2 Transition Functions
12.1.3 Kolmogorov's Equations
12.1.4 Feller and C0 Transition Functions
12.1.5 Path Functions of Markov Processes
12.1.6 Stopping Times
12.1.7 Definition of Strong Markov Processes
12.1.8 Strong Markov Property and Uniform Stochastic Continuity
12.2 Feller Semigroups and Transition Functions
12.2.1 Definition of Feller Semigroups
12.2.2 Characterization of Feller Semigroups in Terms of Transition Functions
12.3 The Hille–Yosida Theory of Feller Semigroups
12.3.1 Generation Theorems for Feller Semigroups
12.3.2 Generation Theorems for Feller Semigroups in Terms of Maximum Principles
12.4 Infinitesimal Generator of Feller Semigroups on a Bounded Domain (i)
12.5 Infinitesimal Generator of Feller Semigroups on a Bounded Domain (ii)
12.6 Feller Semigroups and Boundary Value Problems
12.7 Notes and Comments
13 L2 Approach to the Construction of Feller Semigroups
13.1 Statements of Main Results
13.2 Proof of Theorem 13.1
13.2.1 Proof of Theorem 13.5
13.3 Proof of Theorem 13.3
13.3.1 Proof of Theorem 13.15
13.4 The Degenerate Diffusion Operator Case
13.4.1 The Regular Boundary Case
13.4.2 The Totally Characteristic Case
13.5 Notes and Comments
14 Concluding Remarks
Appendix A Brief Introduction to the Potential Theoretic Approach
A.1 Hölder Continuity and Hölder Spaces
A.2 Interior Estimates for Harmonic Functions
A.3 Hölder Regularity for the Newtonian Potential
A.4 Hölder Estimates for the Second Derivatives
A.5 Hölder Estimates at the Boundary
A.6 Notes and Comments
Appendix References
Index
