一本介绍 Markov Processes 的书籍,希望对读者有所帮助.

目录如下:

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 Lp-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 Processes
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 Pn-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 Lp-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 Lp-sub-Markovian Semigroups 252
5.4 Markov Processes Associated with an Lp-sub-Markovian Semigroup
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