Chapter 1 Markov Process

1.1．Markov Property

1.2．Transition Function

1.3．Optional Times

1.4．Martingale Theorems

1.5．Progressive Measurability and the Section Theorem

Exercises

Notes on Chapter 1

Chapter 2 Basic Properties

2.1．Martingale Connection

2.2．Feller Process

Exercises

2.3．Strong Markov Property and Right Continuity of Fields

Exercises

2.4．Moderate Markov Property and Quasi Left Continuity

Exercises

Notes on Chapter 2

Chapter 3 Hunt Process

3.1．Defining Properties

Exercises

3.2．Analysis of Excessive Functions

Exercises

3.3．Hitting Times

3.4．Balayage and Fundamental Structure

Exercises

3.5．Fine Properties

Exercises

3.6．Decreasing Limits

Exercises

3.7．Recurrence and Transience

Exercises

3.8．Hypothesis(B)

Exercises

Notes on Chapter 3

Chapter 4 Brownian Motion

4.1．Spatial Homogeneity

Exercises

4.2．Preliminary Properties of Brownian Motion

Exercises

4.3．Harmonic Function

Exercises

4.4．Dirichlet Problem

Exercises

4.5．Superharmonic Function and Supermartingale

Exercises

4.6．The Role of the Laplacian

Exercises

4.7．The Feynman-Kac Functional and the Schr？dinger Equation

Exercises

Notes on Chapter 4

Chapter 5 Potential Developments

5.1．Quitting Time and Equilibrium Measure

Exercises

5.2．Some Principles of Potential Theory

Exercises

Notes on Chapter 5

Chapter 6 Generalities

6.1 Essential Limits

6.2 Penetration Times

6.3 General Theory

Exercises

Notes on Chapter 6

Chapter 7 Markov Chains：a Fireside Chat

7.1 Basic Examples

Nores on Chapter 7

Chapter 8 Ray Processes

8.1 Ray Resolvents and Semigroups

8.2 Branching Points

8.3 The Rav Processes

8.4 Jumps and Branching Points

8.5 Martingales on the Ray Space

8.6 A Feller Property of Px

8.7 Jumps Without Branching Points

8.8 Bounded Entrance Laws

8.9 Regular Supermedian Functions

8.10 Ray-Knight Compactifications：Why Every Markov Process is a Ray Process at Heart

8.11 Useless Sets

8.12 Hunt Processes and Standard Processes

8.13 Separation and Supermedian Functions

8.14 Examples

Exercises

Notes on Chapter 8

Chapter 9 Application to Markov Chains

9.1 Compactifications of Markov Chains

9.2 Elementary Path Properties of Markov Chains

9.3 Stable and Instantaneous States

9.4 A Second Look at the Examples of Chapter 7

Exercises

Notes on Chapter 9

Chapter 10 Time Reversal

10.1 The Loose Transition Function

10.2 Improving the Resolvent

10.3 Proof of Theorem 10.1

10.4 Removing Hypotheses(H1)and(H2)

Notes on Chapter 10

Chapter 11 h-Transforms

11.1 Branching Points

11.2 h-Transforms

11.3 Construction of the h-Processes

11.4 Minimal Excessive Functions and the Invariant Field

11.5 Last Exit and Co-optional Times

11.6 Reversing h-Transforms

Exercises

Nores on Chapter 11

Chapter 12 Death and Transfiguration：A Fireside Chat

Exercises

Notes on Chapter 12

Chapter 13 Processes in Duality

13.1 Formal Duality

13.2 Dual Processes

13.3 Excessive Measures

13.4 Simple Time Reversal

13.5 The Moderate Markov Property

13.6 Dual Quantities

13.7 Small Sets and Regular Points

13.8 Duality and h-Transforms

Exercises

13.9 Reversal From a Random Time

13.10 Xζ-：Limits at the Lifetime

13.11 Balayage and Potentials of Measures

13.12 The Interior Reduite of a Function

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

13.14 Fine Symmetry

13.15 Capacities and Last Exit Times

Exercises

Notes on Chapter 13

Chapter 14 The Martin Boundary

14.1 Hypotheses

14.2 The Martin Kernel and the Martin Space

14.3 Minimal Points and Boundary Limits

14.4 The Martin Representation

14.5 Applications

14.6 The Martin Boundary for Brownian Motion

14.7 The Dirichlet Problem in the Martin Space

Exercises

Notes on Chapter 14

Chapter 15 The Basis of Duality：A Fireside Chat

15.1 Duality Measures

15.2 The Cofine Topology

Notes on Chapter 15

Bibliography

Index