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[资源] Pseudo Differential Operators & Markov Processes

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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2016-09-14 19:42:00, 17.74 M