Chapter 1 Markov Process 1
1.1．Markov Property 1
1.2．Transition Function 6
1.3．Optional Times 12
1.4．Martingale Theorems 24
1.5．Progressive Measurability and the Section Theorem 37
Exercises 43
Notes on Chapter 1 44
Chapter 2 Basic Properties 45
2.1．Martingale Connection 45
2.2．Feller Process 48
Exercises 55
2.3．Strong Markov Property and Right Continuity of Fields 56
Exercises 65
2.4．Moderate Markov Property and Quasi Left Continuity 66
Exercises 73
Notes on Chapter 2 73
Chapter 3 Hunt Process 75
3.1．Defining Properties 75
Exercises 78
3.2．Analysis of Excessive Functions 80
Exercises 87
3.3．Hitting Times 87
3.4．Balayage and Fundamental Structure 96
Exercises 105
3.5．Fine Properties 106
Exercises 115
3.6．Decreasing Limits 116
Exercises 122
3.7．Recurrence and Transience 122
Exercises 130
3.8．Hypothesis(B) 130
Exercises 135
Notes on Chapter 3 135
Chapter 4 Brownian Motion 137
4.1．Spatial Homogeneity 137
Exercises 143
4.2．Preliminary Properties of Brownian Motion 144
Exercises 152
4.3．Harmonic Function 154
Exercises 160
4.4．Dirichlet Problem 162
Exercises 173
4.5．Superharmonic Function and Supermartingale 174
Exercises 187
4.6．The Role of the Laplacian 189
Exercises 198
4.7．The Feynman-Kac Functional and the Schr？dinger Equation 199
Exercises 205
Notes on Chapter 4 206
Chapter 5 Potential Developments 208
5.1．Quitting Time and Equilibrium Measure 208
Exercises 217
5.2．Some Principles of Potential Theory 218
Exercises 229
Notes on Chapter 5 232
Chapter 6 Generalities 233
6.1 Essential Limits 233
6.2 Penetration Times 237
6.3 General Theory 238
Exercises 242
Notes on Chapter 6 243
Chapter 7 Markov Chains：a Fireside Chat 244
7.1 Basic Examples 244
Nores on Chapter 7 249
Chapter 8 Ray Processes 250
8.1 Ray Resolvents and Semigroups 250
8.2 Branching Points 254
8.3 The Rav Processes 255
8.4 Jumps and Branching Points 258
8.5 Martingales on the Ray Space 259
8.6 A Feller Property of Px 261
8.7 Jumps Without Branching Points 263
8.8 Bounded Entrance Laws 265
8.9 Regular Supermedian Functions 265
8.10 Ray-Knight Compactifications：Why Every Markov Process is a Ray Process at Heart 268
8.11 Useless Sets 274
8.12 Hunt Processes and Standard Processes 276
8.13 Separation and Supermedian Functions 279
8.14 Examples 286
Exercises 288
Notes on Chapter 8 290
Chapter 9 Application to Markov Chains 291
9.1 Compactifications of Markov Chains 292
9.2 Elementary Path Properties of Markov Chains 293
9.3 Stable and Instantaneous States 295
9.4 A Second Look at the Examples of Chapter 7 297
Exercises 301
Notes on Chapter 9 302
Chapter 10 Time Reversal 303
10.1 The Loose Transition Function 307
10.2 Improving the Resolvent 311
10.3 Proof of Theorem 10.1 316
10.4 Removing Hypotheses(H1)and(H2) 316
Notes on Chapter 10 317
Chapter 11 h-Transforms 320
11.1 Branching Points 321
11.2 h-Transforms 321
11.3 Construction of the h-Processes 324
11.4 Minimal Excessive Functions and the Invariant Field 326
11.5 Last Exit and Co-optional Times 329
11.6 Reversing h-Transforms 332
Exercises 334
Nores on Chapter 11 334
Chapter 12 Death and Transfiguration：A Fireside Chat 336
Exercises 341
Notes on Chapter 12 341
Chapter 13 Processes in Duality 342
13.1 Formal Duality 343
13.2 Dual Processes 347
13.3 Excessive Measures 349
13.4 Simple Time Reversal 351
13.5 The Moderate Markov Property 354
13.6 Dual Quantities 356
13.7 Small Sets and Regular Points 361
13.8 Duality and h-Transforms 364
Exercises 365
13.9 Reversal From a Random Time 365
13.10 Xζ-：Limits at the Lifetime 371
13.11 Balayage and Potentials of Measures 375
13.12 The Interior Reduite of a Function 377
13.13 Quasi-left-continuity,Hypothesis(B),and Reduites 384
13.14 Fine Symmetry 388
13.15 Capacities and Last Exit Times 394
Exercises 395
Notes on Chapter 13 396
Chapter 14 The Martin Boundary 398
14.1 Hypotheses 398
14.2 The Martin Kernel and the Martin Space 399
14.3 Minimal Points and Boundary Limits 403
14.4 The Martin Representation 404
14.5 Applications 408
14.6 The Martin Boundary for Brownian Motion 410
14.7 The Dirichlet Problem in the Martin Space 411
Exercises 413
Notes on Chapter 14 414
Chapter 15 The Basis of Duality：A Fireside Chat 416
15.1 Duality Measures 416
15.2 The Cofine Topology 417
Notes on Chapter 15 420
Bibliography 421
Index 426