Markov Processes and Potential Theory (eBook)

Front Cover 1
Markov Processes and Potential Theory 6
Copyright Page 7
Contents 10
Preface 8
Chapter 0. Preliminaries 12
1. Notation 12
2. The Monotone Class Theorem 16
3. Topological Spaces 19
Chapter I. Markov Processes 22
1. General Definitions 22
2. Transition Functions 25
3. General Definitions Continued 31
4. Equivalent Processes 35
5. The Measures Pµ 36
6. Stopping Times 41
7. Stopping Times for Markov Processes 47
8. The Strong Markov Property 48
9. Standard Processes 55
10. Measurability of Hitting Times 63
11. Further Properties of Hitting Times 72
12. Regular Step Processes 74
Chapter ll. Excessive Functions 80
1. Introduction 80
2. Excessive Functions 83
3. Exceptional Sets 90
4. The Fine Topology 95
5. Alternative Characterization of Excessive Functions 100
Chapter lll. Multiplicative Functionals and Subprocesses 108
1. Multiplicative Functionals 108
2. Subordinate Semigroups 112
3. Subprocesses 116
4. Resolvents and Strong Multiplicative Functionals 124
5. Excessive Functions 136
6. A Theorem of Hunt 147
Chapter IV. Additive Functionals and Their Potentials 158
1. Additive Functionals 159
2. Potentials of Additive Functionals 165
3. Potentials of Continuous Additive Functionals 171
4. Potentials of Natural Additive Functionals 181
5. Classification of Excessive Functions 197
Chapter V. Further Properties of Continuous Additive Functionals 207
1. Reference Measures 207
2. Continuous Additive Functionals 216
3. Fine Supports and Local Times 223
4. Balayage of Additive Functionals 239
5. Processes with Identical Hitting Distributions 244
Chapter VI. Dual Processes and Potential Theory 263
1. Dual Processes 263
2. Potentials of Measures 276
3. Potentials of Additive Functionals 290
4. Capacity and Related Topics 294
Notes and Comments 306
Bibliography 316
Index of Notation 322
Subject Index 323