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Chapter 1 Markov Process1

1.1．Markov Property1

1.2．Transition Function6

1.3．Optional Times12

1.4．Martingale Theorems24

1.5．Progressive Measurability and the Section Theorem37

Exercises43

Notes on Chapter 144

Chapter 2 Basic Properties45

2.1．Martingale Connection45

2.2．Feller Process48

Exercises55

2.3．Strong Markov Property and Right Continuity of Fields56

Exercises65

2.4．Moderate Markov Property and Quasi Left Continuity66

Exercises73

Notes on Chapter 273

Chapter 3 Hunt Process75

3.1．Defining Properties75

Exercises78

3.2．Analysis of Excessive Functions80

Exercises87

3.3．Hitting Times87

3.4．Balayage and Fundamental Structure96

Exercises105

3.5．Fine Properties106

Exercises115

3.6．Decreasing Limits116

Exercises122

3.7．Recurrence and Transience122

Exercises130

3.8．Hypothesis(B)130

Exercises135

Notes on Chapter 3135

Chapter 4 Brownian Motion137

4.1．Spatial Homogeneity137

Exercises143

4.2．Preliminary Properties of Brownian Motion144

Exercises152

4.3．Harmonic Function154

Exercises160

4.4．Dirichlet Problem162

Exercises173

4.5．Superharmonic Function and Supermartingale174

Exercises187

4.6．The Role of the Laplian189

Exercises198

4.7．The Feynman-K Functional and the Schr？dinger Equation199

Exercises205

Notes on Chapter 4206

Chapter 5 Potential Developments208

5.1．Quitting Time and Equilibrium Measure208

Exercises217

5.2．Some Principles of Potential Theory218

Exercises229

Notes on Chapter 5232

Chapter 6 Generalities233

6.1 Essential Limits233

6.2 Peration Times237

6.3 General Theory238

Exercises242

Notes on Chapter 6243

Chapter 7 Markov Chains：a Fireside Chat244

7.1 Basic Examples244

Nores on Chapter 7249

Chapter 8 Ray Processes250

8.1 Ray Resolvents and Semigroups250

8.2 Branching Points254

8.3 The Rav Processes255

8.4 Jumps and Branching Points258

8.5 Martingales on the Ray Spe259

8.6 A Feller Property of Px261

8.7 Jumps Without Branching Points263

8.8 Bounded Entrance Laws265

8.9 Regular Supermedian Functions265

8.10 Ray-Knight Comptifications：Why Every Markov Process is a Ray Process at Heart268

8.11 Useless Sets274

8.12 Hunt Processes and Standard Processes276

8.13 Separation and Supermedian Functions279

8.14 Examples286

Exercises288

Notes on Chapter 8290

Chapter 9 Application to Markov Chains291

9.1 Comptifications of Markov Chains292

9.2 Elementary Path Properties of Markov Chains293

9.3 Stable and Instantaneous States295

9.4 A Second Look at the Examples of Chapter 7297

Exercises301

Notes on Chapter 9302

Chapter 10 Time Reversal303

10.1 The Loose Transition Function307

10.2 Improving the Resolvent311

10.3 Proof of Theorem 10.1316

10.4 Removing Hypotheses(H1)and(H2)316

Notes on Chapter 10317

Chapter 11 h-Transforms320

11.1 Branching Points321

11.2 h-Transforms321

11.3 Construction of the h-Processes324

11.4 Minimal Excessive Functions and the Invariant Field326

11.5 Last Exit and Co-optional Times329

11.6 Reversing h-Transforms332

Exercises334

Nores on Chapter 11334

Chapter 12 Death and Transfiguration：A Fireside Chat336

Exercises341

Notes on Chapter 12341

Chapter 13 Processes in Duality342

13.1 Formal Duality343

13.2 Dual Processes347

13.3 Excessive Measures349

13.4 Simple Time Reversal351

13.5 The Moderate Markov Property354

13.6 Dual Quantities356

13.7 Small Sets and Regular Points361

13.8 Duality and h-Transforms364

Exercises365

13.9 Reversal From a Random Time365

13.10 Xζ-：Limits at the Lifetime371

13.11 Balayage and Potentials of Measures375

13.12 The Interior Rite of a Function377

13.13 Quasi-left-continuity,Hypothesis(B),and Rites384

13.14 Fine Symmetry388

13.15 Capities and Last Exit Times394

Exercises395

Notes on Chapter 13396

Chapter 14 The Martin Boundary398

14.1 Hypotheses398

14.2 The Martin Kernel and the Martin Spe399

14.3 Minimal Points and Boundary Limits403

14.4 The Martin Representation404

14.5 Applications408

14.6 The Martin Boundary for Brownian Motion410

14.7 The Dirichlet Problem in the Martin Spe411

Exercises413

Notes on Chapter 14414

Chapter 15 The Basis of Duality：A Fireside Chat416

15.1 Duality Measures416

15.2 The Cofine Topology417

Notes on Chapter 15420

Bibliography421

Index426