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

Description

This 3rd edition provides an insight into the mathematical crossroads formed by functional analysis (the macroscopic approach), partial differential equations (the mesoscopic approach) and probability (the microscopic approach) via the mathematics needed for the hard parts of Markov processes. It brings these three fields of analysis together, providing a comprehensive study of Markov processes from a broad perspective. The material is carefully and effectively explained, resulting in a surprisingly readable account of the subject. The main focus is on a powerful method for future research in elliptic boundary value problems and Markov processes via semigroups, the Boutet de Monvel calculus. A broad spectrum of readers will easily appreciate the stochastic intuition that this edition conveys. In fact, the book will provide a solid foundation for both researchers and graduate students in pure and applied mathematics interested in functional analysis, partial differential equations, Markov processes and the theory of pseudo-differential operators, a modern version of the classical potential theory. 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