Boundary Value Problems and Markov Processes: Functional Analysis Methods for Markov Processes

Preface to the Third Edition
Preface to the Second Edition
Contents
1 Introduction and Main Results
	1.1 Historical Perspective of Feller\'s Approach to Brownian Motion
	1.2 Formulation of the Problem and Statement of Main Results
		1.2.1 The Differential Operator Case
			(I) Analytic Semigroups in the Lp Topology
			(II) Analytic Semigroups in the Topology of Uniform Convergence and Feller Semigroups
		1.2.2 The Integro-Differential Operator Case
			(I) Unique Solvability Theorems for Waldenfels Integro-Differential Operators
			(II) Analytic Semigroups in the Lp Topology
			(III) Analytic Semigroups in the Topology of Uniform Convergence and Feller Semigroups
	1.3 Summary of the Contents
	1.4 An Overview of Main Theorems
	1.5 Notes and Comments
Part I Analytic and Feller Semigroups and Markov Processes
	2 Analytic Semigroups
		2.1 Analytic Semigroups via the Cauchy Integral
		2.2 Generation Theorem for Analytic Semigroups
		2.3 Remark on the Resolvent Estimate (2.2)
		2.4 Notes and Comments
	3 Markov Processes and Feller Semigroups
		3.1 Continuous Functions and Measures
			3.1.1 Space of Continuous Functions
			3.1.2 Space of Signed Measures
			3.1.3 The Riesz–Markov Representation Theorem
			3.1.4 Weak Convergence of Measures
		3.2 Elements of Markov Processes
			3.2.1 Definition of Markov Processes
			3.2.2 Markov Processes and Markov Transition Functions
			3.2.3 Path Functions of Markov Processes
			3.2.4 Stopping Times
			3.2.5 Definition of Strong Markov Processes
			3.2.6 Strong Markov Property and Uniform Stochastic Continuity
		3.3 Markov Transition Functions and Feller Semigroups
		3.4 Generation Theorems for Feller Semigroups
		3.5 Reflecting Diffusion
		3.6 Local Time on the Boundary for the Reflecting Diffusion
		3.7 Notes and Comments
Part II Pseudo-Differential Operators and Elliptic Boundary Value Problems
	4 Lp Theory of Pseudo-Differential Operators
		4.1 Function Spaces
			4.1.1 Hölder Spaces
			4.1.2 Lp Spaces
			4.1.3 Fourier Transforms
			4.1.4 Tempered Distributions
			4.1.5 Sobolev Spaces
			4.1.6 Besov Spaces
			4.1.7 General Sobolev and Besov Spaces
			4.1.8 Sobolev\'s Imbedding Theorems
			4.1.9 The Rellich–Kondrachov theorem
		4.2 Seeley\'s Extension Theorem
		4.3 Trace Theorems
			4.3.1 Sectional Traces
			4.3.2 Jump Formulas
		4.4 Fourier Integral Operators
			4.4.1 Symbol Classes
			4.4.2 Phase Functions
			4.4.3 Oscillatory Integrals
			4.4.4 Definitions and Basic Properties of Fourier Integral Operators
		4.5 Pseudo-Differential Operators
			4.5.1 Definitions and Basic Properties of Pseudo-Differential Operators
			4.5.2 Lp Boundedness of Pseudo-Differential Operators
			4.5.3 Pseudo-Differential Operators on a Manifold
			4.5.4 Hypoelliptic Pseudo-Differential Operators
			4.5.5 Distribution Kernel of Pseudo-Differential Operators
		4.6 Elliptic Pseudo-differential Operators and their Indices
			4.6.1 Pseudo-Differential Operators on Sobolev Spaces
			4.6.2 The Index of an Elliptic Pseudo-Differential Operator
		4.7 Functional Calculus for the Laplacian via the Heat Kernel
		4.8 Notes and Comments
	5 Boutet de Monvel Calculus
		5.1 The Spaces H, H+ and H-
		5.2 Transmission Property of Pseudo-Differential Operators
		5.3 Trace, Potential and Singular Green Operators on Rn+
			5.3.1 Potential Operators on Rn+
			5.3.2 Trace Operators on Rn+
			5.3.3 Singular Green Operators on Rn+
			5.3.4 Boundary Operators on Rn-1
		5.4 Historical Perspective of the Wiener–Hopf Technique
		5.5 Notes and Comments
	6 Lp Theory of Elliptic Boundary Value Problems
		6.1 Classical Potentials and Pseudo-Differential Operators
			6.1.1 Single and Double Layer Potentials
			6.1.2 The Green Representation Formula
			6.1.3 Surface and Volume Potentials
		6.2 Dirichlet Problem
		6.3 Formulation of the Boundary Value Problem
		6.4 Special Reduction to the Boundary
		6.5 Boundary Value Problems via Boutet de Monvel Calculus
			6.5.1 Boundary Space Bs-1-1/p,p(∂Ω)
			6.5.2 Index Formula of Agranovič–Dynin Type
		6.6 Dirichlet-to-Neumann Operator and Reflecting Diffusion
		6.7 Spectral Analysis of the Dirichlet Eigenvalue Problem
			6.7.1 Unique Solvability of the Dirichlet Problem
			6.7.2 A Characterization of the Resolvent set of the Dirichlet Problem
		6.8 Notes and Comments
Part III Analytic Semigroups in Lp Sobolev Spaces
	7 Proof of Theorem 1.2
		7.1 Boundary Value Problem with Spectral Parameter
		7.2 Proof of the A Priori Estimate (1.7)
		7.3 Notes and Comments
	8 A Priori Estimates
		8.1 A Priori Estimates via Agmon\'s Method
		8.2 Notes and Comments
	9 Proof of Theorem 1.4
		9.1 Proof of Theorem 1.4, Part (i)
			9.1.1 Proof of Proposition 9.2
		9.2 Proof of Theorem 1.4, Part (ii)
		9.3 Notes and Comments
Part IV Waldenfels Operators, Boundary Operators and Maximum Principles
	10 Elliptic Waldenfels Operators and Maximum Principles
		10.1 Borel Kernels and Maximum Principles
			10.1.1 Linear Operators having Positive Borel Kernel
			10.1.2 Borel Kernels and Pseudo-Differential Operators
		10.2 Maximum Principles for Elliptic Waldenfels Operators
			10.2.1 Weak Maximum Principle
			10.2.2 Strong Maximum Principle
			10.2.3 Hopf\'s Boundary Point Lemma
		10.3 Notes and Comments
	11 Boundary Operators and Boundary Maximum Principles
		11.1 Ventcel\'–Lévy Boundary Operators
		11.2 Positive Boundary Maximum Principles
			11.2.1 Boundary Maximum Principles for Ventcel\'–Lévy operators
			11.2.2 Boundary Maximum Principles for Ventcel\' operators
				Proof of Theorem 11.4, Part (i)
				Proof of Theorem 11.4, Part (ii)
		11.3 Notes and Comments
Part V Feller Semigroups for Elliptic Waldenfels Operators
	12 Proof of Theorem 1.5 - Part (i) -
		12.1 Space C0(D M)
		12.2 Proof of Part (i) of Theorem 1.5
		12.3 Notes and Comments
	13 Proofs of Theorem 1.5, Part (ii) and Theorem 1.6
		13.1 General Existence Theorem for Feller Semigroups
		13.2 Feller Semigroups with Reflecting Barrier
		13.3 Proof of Theorem 1.6
		13.4 Proof of Part (ii) of Theorem 1.5
		13.5 Notes and Comments
	14 Proofs of Theorems 1.8, 1.9, 1.10 and 1.11
		14.1 Existence and Uniqueness Theorem in Hölder Spaces
		14.2 Proof of Theorem 1.8
		14.3 Proof of Theorem 1.9
		14.4 Proof of Theorem 1.10
		14.5 Proof of Theorem 1.11
		14.6 Minimal Closed Extension W
		14.7 Notes and Comments
	15 Path Functions of Markov Processes via Semigroup Theory
		15.1 Basic Definitions and Properties of Markov Processes
		15.2 Path-Continuity of Markov Processes
		15.3 Path-Continuity of Feller Semigroups
		15.4 Examples of Multi-dimensional Diffusion Processes
			15.4.1 The Neumann Case
			15.4.2 The Robin Case
			15.4.3 The Oblique Derivative Case
		15.5 Notes and Comments
Part VI Concluding Remarks
	16 The State-of-the-Art of Generation Theorems for Feller Semigroups
		16.1 Formulation of the Problem
		16.2 Statement of Main Results
			16.2.1 The Transversal Case
			16.2.2 The Non-Transversal Case
			16.2.3 The Lower Order Case
		16.3 Notes and Comments
	References
	Index
References
Index