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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 Chapter4 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 SchrOdinger 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 Notes on Chapter 7 249 Chapter 8 Ray Processes 250 8.1 Ray Resolvents and Semigroups 250 8.2 Branching Points 254 8.3 The Ray 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 (HI) 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 Notes 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 Xc_: 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