Functional Analytic Techniques for Diffusion Processes

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