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内容简介

　　This book consists of two parts， to be called Part I and Part n. Part I，Chapters l through 5， is essentially a new edition of Kai Lai Chung's Lec-tures from Markov Processes to Brownian Motion （1982）. He has corrected a number of misprints in the original edition， and has inserted a few references and remarks， of which he says， "The latter must be regarded as randomly selected since twenty-some years is a long time to retrace steps..." This part introduces strong Markov processes and their potential theory. In particular，it studies Brownian motion， and shows how it generates classical potential theory.

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目录

Chapter 1 Markov Ptocess
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 I

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. FineProperties
Exercises
3.6.Decreasing Limits
Exercises
3.7.Recurrence and Transience
Exercises
3.8.Hypothesis (B)
Exerases
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
Exerases
4.4.Dirichlet Problem
Exercises
4.5.Superharmonic Function and Supermartingale
Exerases
4.6.The Role of the Laplacian
Exercises
4.7. The Feynman-Kac Functional and the Schrodinger Equation
Exe工ases
Notes on Chapter 4

Chapter 5 Potential Developments
5.1 Quitting Time and Equilibrium Measure
Exercises
5.2.Some Princip les of Potential Theory
Exerases
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
Notes on Chapter 7

Chapter 8 Ray Processes
8.1 Ray Resolvents and Semigroups
8.2Branching Points
Chapter9
Application to Markov Chains
……
Chapter 10 Time Reversal
Chapter 11 h-Transforms
Chapter 12 Death and Transfiguration: A Fireside Chat
Chapter 13 Processes in Duality
Chapter 14 The Martin Boundary
Chapter 15 The Basis of Duality: A Fireside Chat
Bibliography
Index

精彩书摘

　　This book consists of two parts， to be called Part I and Part n. Part I， Chapters l through 5， is essentially a new edition of Kai Lai Chung's Lec-tures from Markov Processes to Brownian Motion （1982）. He has corrected a number of misprints in the original edition， and has inserted a few references and remarks， of which he says， "The latter must be regarded as randomly selected since twenty-some years is a long time to retrace steps . . ." This part introduces strong Markov processes and their potential theory. In particular，it studies Brownian motion， and shows how it generates classical potential theory.
　　Part II， Chapters 6 through 15， began life as a set of notes for a series of lectures on time reversal and duality given at the University of Paris. I originally planned to add the essential parts of these notes to this edition to show how the reversal of time-the retracing of steps-explained so much about Markov processes and their potential theory. But like many others， I learned that the inessential parts of a cherished manuscript form at most a fuzzy empty set， while the essential parts include everything that should have been in the original， even if it wasn't. In short， this， like Topsy， just grow'd.
　　Indeed， reversal and duality are best understood in light of Ray processes and the Ray-Knight compactification. But it is fitting that a study of symmetry be symmetrical itself， so once I had included the Ray compactification， I had to include its mirror image， the Martin boundary. This was followed by a host of examples， remarks and theorems to show how these new ideas influence the theory and practice developed in the first part. The result was the present Part II.
　　In a sense， Part II deals with the same subjects as Pan I， but more narrowly: using Part I for a general understanding， we are free to focus on the effects of time reversal， duality， and time-symmetry on potential theory. Certain theo-rems in Part I are re-proved in Part II under slightly weaker hypotheses. This . is not because I want to generalize the theorems， but because I want to show
　　them in a different light: the proofs in Part II are quite different from those of Part I.
　　……

前言/序言

经典数学教材（影印版）：马尔科夫过程、布朗运动和时间对称（第2版） 电子书 下载 mobi epub pdf txt

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