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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