钟开来著的《马尔科夫过程布朗运动和时间对称(第2版)》是基于过去20年间的几份讲义一些部分创作而成，其原型是作者于1970年春天的一学期讲义。书中旨在将马尔科夫过程的一些最好的特性，特别地，用最少的预备知识和技巧讲述了布朗运动。这新的版本新增加了9章，包括新的练习、参考资料和原来版本的多处修订。目次：马尔科夫过程；基本上性质；Hunt过程；布朗运动；势发展；综述；马尔科夫链；放射过程；马尔科夫链的应用；时间逆转；h－变换；灭亡与变形；对偶过程；Martin边界。读者对象：数学、物理专业的研究生、老师和相关的科研人员。

Preface to the New Edition

Prefaee to the First Edition

Chapter 1

Markov Process

1.1. Markov Property

1.2. Transition Funotion

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

Exerclses

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

Chapter4

Brownian Motion

4.1. Spatial Homogeneity

Exercises

4.2. Preliminary Properties of Brownian Motion

Exercises

4.3. Harmonie Function

Exercises

4.4. Dirichlet Problem

Exetcises

4.5. Superharmomc Function and Supermartingale

Exercises

4.6. The Role ofthe Laplacian

Exercises

4.7. The Feynman-Kac Functional and the Schrodinger Equation

Exercises

Notes on Chapter 4

Chapter 5

Potential Developments

5.1. QuittingTime and Equilibrium Measure

Exercises

5.2. Some Principles of Potential Theory

Exercises

Notes on Chapter 5

Chapter 6

Generalities

6.1. Essenfial Limits

6.2. Penetration Tiriles

6.3. General Theory

Exemises

Notes on Chapter 6

Chapter7

Markov Chains：a Fireside Chat

7.1 Basic Examples

Nores on Chapter 7

Chapter8

Ray Processes

8.1. Ray Resolvents and Semigroups

8.2. Branching Points

8.3. The Ray Processes

8.4. Jumps and Branching Points

8.5. Martingales on the Ray Space

8.6. A FetierProperty of Px

8.7. Jumps Without Branching Points

8.8. Bounded En~ance Laws

8.9. Regutar Snpetmedian Functions

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

 Process at Hcart

8.11. Useless Sets

8.12. Hunt Processes and Standard Processes

8.13. Separation and Supermedian Funcfions

8 14. Examples

 Exercises

 Nores on Chapter 8

Chapter9

Application to Markov Chains

9.1. Compactifications of Markov Chaias

9.2. Elementary Path Properties of Markov chaias

9 3. Stable and Instantaneous States

9.4. A Second Look at the Examples ofChapter7

Exercises

Notes on Chapter8

Chapter 10

Time Reversal

10.1. the Loose nansition Function

10.2. Improving the Resolvent

10.3. PtoofofTheorem 10.1

10.4. Removing Hypotheses（H1）and（H2）

 Nores on Chapter 10

Chapter 11

h-Transforms

11.1. Branching Points

11.2. h-Transforms

11.3. Construction ofthe 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 Mensures

13.4. Simple Time Reversal

13.5. The Moderate Markov Property

13.6. Dual Quantities

13.7. SmalJ Sets and Regular Points

13.8. Duality and h-Transforms

 Exercises

13.9. Reversal Ftom a Random Time

13.10. X_：Limits at the Lifetime

13.11. Balayage and Potentials of Measures

13.12. The Interior Reduite of a Function

13.13. Quasi-left-confinuity，Hypothesis（B），and Reduites

13.14. Fine Symmetry

13.15. Capacities and Last Exit Times

  Exercises

  Nores on Chapter 13

Chapter 14

The Martin Boundary

14.1. Hypotheses

14.2. The Martin Kerneland 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 Morion

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