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Pseudo Differential Operators & Markov Processes
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Contents Preface ix Notation xi General Notation xi Functions and Distributions xiii Measures and Integrals xiv Spaces of Functions, Measures and Distributions xiv Some Families of Functions xvi Norms, Scalar Products and Seminorms xvii Notation from Functional Analysis, Operators xvii Notations related to Pseudo-Differential Operators xix Notations related to Potential Theory and Harmonic Analysis . . . . xix Notation from Probability Theory xx Stochastic Processes and related Notations xxi Notations related to cadlag-Functions xxii Notations related to Hunt Processes and L p -Markovian Semigroups . xxiii Some more Special Notations xxiii Introduction: Pseudo-Differential Operators and Markov Processes xxv III Markov Processes and Applications 1 1 Introduction 3 2 Essentials from Probability Theory 9 2.1 Some Remarks on cr-Fields and Measures 9 vj Contents 2.2 Some Integration Theory 17 2.3 Measure and Topology 22 2.4 Independence 24 2.5 Conditional Expectation 27 2.6 Martingales and Stopping Times 33 2.7 Some Remarks to Stochastic Processes 41 3 Feller Processes 45 3.1 Projective Limits and Canonical Processes 46 3.2 Semigroups of Kernels, Transition Functions and Canonical Pro- cesses 51 3.3 A First Encounter with Sample Paths and Cadlag-Functions . . 71 3.4 Markov Processes and Feller Processes 80 3.5 The Shift Operator and the Strong Markov Property 96 3.6 The Martingale Problem for Feller Processes 108 3.7 Levy Processes and Translation Invariant Feller Semigroups . . 115 3.8 A Summary of Some Path Properties of Levy Processes . . . . 141 3.9 The Symbol of a Feller Process 146 3.10 Notes to Chapter 3 152 4 The Martingale Problem 157 4.1 Probability Measures on ￡> n ([0,oo)) 158 4.2 Existence Results for the P n -Martingale Problem 174 4.3 A Uniqueness Criterion for the Solvability of the Martingale Problem 191 4.4 A Localization Procedure 198 4.5 On the Well-Posedness of the Martingale Problem 201 4.6 The Martingale Problem and the Feller Property 214 4.7 Notes to Chapter 4 221 5 L p -sub-Markovian Semigroups and Hunt Processes 223 5.1 Why the Theory of Feller Processes is not Sufficient 223 5.2 Hunt Processes 227 5.3 Hunt Processes Associated with L p -sub-Markovian Semigroups 252 5.4 Markov Processes Associated with an L p -sub-Markovian Semi- group 276 5.5 Notes to Chapter 5 279 Contents vii 6 Markov Processes and Potential Theory 283 6.1 Heuristic Links between Markov Processes and Potential Theory 284 6.2 Potential Theoretical Notions and their Probabilistic Counterparts287 6.3 Potential Theory of Levy Processes and more Probabilistic Counterparts 310 6.4 Applications to Markov Processes 319 6.5 The Balayage-Dirichlet Problem 329 6.6 Notes to Chapter 6 351 7 Selected Applications and Extensions 353 7.1 Fractional Derivatives as Generators of Markov Processes . . . 354 7.2 Remarks on Markov Processes with State Spaces Having a Boundary 368 7.3 Making Parameters State Space Dependent 385 7.4 Remarks on Stochastic Spectral Analysis 389 7.5 Function Spaces Associated with a Continuous Negative Definite Function 393 7.6 Notes to Chapter 7 400 A Parametrix Construction for Evolution Equations 403 B A Parameter Dependent Extension of Hoh's Calculus 407 C On Roth's Method for Constructing Feller Semigroups 411 D More Continuous Negative Definite Functions 415 E More (Complete) Bernstein Functions 419 Corrections to Volume I 422 Corrections to Volume II 423 Changes in the Bibliography of Volume II 424 Bibliography 425 Author Index 457 Subject Index 463 |
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