Markov Processes, Brownian Motion, and Time Symmetry

by Chung, Kai Lai; WALSH, JOHN B.

• ISBN13: 9780387220260
• ISBN10: 0387220267
• Edition: 2nd
• Format: Hardcover
• Copyright: 2005-08-30
• Publisher: Springer Verlag

Summary

From the reviews of the First Edition:"This excellent book is based on several sets of lecture notes written over a decade and has its origin in a one-semester course given by the author at the ETH, Zürich, in the spring of 1970. The author's aim was to present some of the best features of Markov processes and, in particular, of Brownian motion with a minimum of prerequisites and technicalities. The reader who becomes acquainted with the volume cannot but agree with the reviewer that the author was very successful in accomplishing this goal'¦The volume is very useful for people who wish to learn Markov processes but it seems to the reviewer that it is also of great interest to specialists in this area who could derive much stimulus from it. One can be convinced that it will receive wide circulation." (Mathematical Reviews)This new edition contains 9 new chapters which include new exercises, references, and multiple corrections throughout the original text.

Table of Contents

Preface to the New Edition	ix
Preface to the First Edition	xi
Chapter 1 Markov Process	1	(44)
1.1. Markov Property	1	(5)
1.2. Transition Function	6	(6)
1.3. Optional Times	12	(12)
1.4. Martingale Theorems	24	(13)
1.5. Progressive Measurability and the Section Theorem	37	(7)
Exercises	43	(1)
Notes on Chapter 1	44	(1)
Chapter 2 Basic Properties	45	(30)
2.1. Martingale Connection	45	(3)
2.2. Feller Process	48	(8)
Exercises	55	(1)
2.3. Strong Markov Property and Right Continuity of Fields	56	(10)
Exercises	65	(1)
2.4. Moderate Markov Property and Quasi Left Continuity	66	(7)
Exercises	73	(1)
Notes on Chapter 2	73	(2)
Chapter 3 Hunt Process	75	(62)
3.1. Defining Properties	75	(5)
Exercises	78	(2)
3.2. Analysis of Excessive Functions	80	(7)
Exercises	87	(1)
3.3. Hitting Times	87	(9)
3.4. Balayage and Fundamental Structure	96	(10)
Exercises	105	(1)
3.5. Fine Properties	106	(10)
Exercises	115	(1)
3.6. Decreasing Limits	116	(6)
Exercises	122	(1)
3.7. Recurrence and Transience	122	(8)
Exercises	130	(1)
3.8. Hypothesis (B)	130	(5)
Exercises	135	(1)
Notes on Chapter 3	135	(2)
Chapter 4 Brownian Motion	137	(71)
4.1. Spatial Homogeneity	137	(7)
Exercises	143	(1)
4.2. Preliminary Properties of Brownian Motion	144	(10)
Exercises	152	(2)
4.3. Harmonic Function	154	(8)
Exercises	160	(2)
4.4. Dirichlet Problem	162	(12)
Exercises	173	(1)
4.5. Superharmonic Function and Supermartingale	174	(15)
Exercises	187	(2)
4.6. The Role of the Laplacian	189	(10)
Exercises	198	(1)
4.7. The Feynman-Kac Functional and the Schrödinger Equation	199	(7)
Exercises	205	(1)
Notes on Chapter 4	206	(2)
Chapter 5 Potential Developments	208	(25)
5.1. Quitting Time and Equilibrium Measure	208	(10)
Exercises	217	(1)
5.2. Some Principles of Potential Theory	218	(14)
Exercises	229	(3)
Notes on Chapter 5	232	(1)
Chapter 6 Generalities	233	(11)
6.1 Essential Limits	233	(4)
6.2 Penetration Times	237	(1)
6.3 General Theory	238	(5)
Exercises	242	(1)
Notes on Chapter 6	243	(1)
Chapter 7 Markov Chains: a Fireside Chat	244	(6)
7.1 Basic Examples	244	(5)
Notes on Chapter 7	249	(1)
Chapter 8 Ray Processes	250	(41)
8.1 Ray Resolvents and Semigroups	250	(4)
8.2 Branching Points	254	(1)
8.3 The Ray Processes	255	(3)
8.4 Jumps and Branching Points	258	(1)
8.5 Martingales on the Ray Space	259	(2)
8.6 A Feller Property of PX	261	(2)
8.7 Jumps Without Branching Points	263	(2)
8.8 Bounded Entrance Laws	265	(1)
8.9 Regular Supermedian Functions	265	(3)
8.10 Ray-Knight Compactifications: Why Every Markov Process is a Ray Process at Heart	268	(6)
8.11 Useless Sets	274	(2)
8.12 Hunt Processes and Standard Processes	276	(3)
8.13 Separation and Supermedian Functions	279	(7)
8.14 Examples	286	(4)
Exercises	288	(2)
Notes on Chapter 8	290	(1)
Chapter 9 Application to Markov Chains	291	(12)
9.1 Compactifications of Markov Chains	292	(1)
9.2 Elementary Path Properties of Markov Chains	293	(2)
9.3 Stable and Instantaneous States	295	(2)
9.4 A Second Look at the Examples of Chapter 7	297	(5)
Exercises	301	(1)
Notes on Chapter 9	302	(1)
Chapter 10 Time Reversal	303	(17)
10.1 The Loose Transition Function	307	(4)
10.2 Improving the Resolvent	311	(5)
10.3 Proof of Theorem 10.1	316	(1)
10.4 Removing Hypotheses (H1) and (H2)	316	(1)
Notes on Chapter 10	317	(3)
Chapter 11 h-Transforms	320	(16)
11.1 Branching Points	321	(1)
11.2 h-Transforms	321	(3)
11.3 Construction of the h-Processes	324	(2)
11.4 Minimal Excessive Functions and the Invariant Field	326	(3)
11.5 Last Exit and Co-optional Times	329	(3)
11.6 Reversing h-Transforms	332	(2)
Exercises	334	(1)
Notes on Chapter 11	334	(2)
Chapter 12 Death and Transfiguration: A Fireside Chat	336	(6)
Exercises	341	(1)
Notes on Chapter 12	341	(1)
Chapter 13 Processes in Duality	342	(56)
13.1 Formal Duality	343	(4)
13.2 Dual Processes	347	(2)
13.3 Excessive Measures	349	(2)
13.4 Simple Time Reversal	351	(3)
13.5 The Moderate Markov Property	354	(2)
13.6 Dual Quantities	356	(5)
13.7 Small Sets and Regular Points	361	(3)
13.8 Duality and h-Transforms	364	(1)
Exercises	365	(1)
13.9 Reversal From a Random Time	365	(6)
13.10 Χζ_: Limits at the Lifetime	371	(4)
13.11 Balayage and Potentials of Measures	375	(2)
13.12 The Interior Reduite of a Function	377	(7)
13.13 Quasi-left-continuity, Hypothesis (B), and Reduites	384	(4)
13.14 Fine Symmetry	388	(6)
13.15 Capacities and Last Exit Times	394	(2)
Exercises	395	(1)
Notes on Chapter 13	396	(2)
Chapter 14 The Martin Boundary	398	(18)
14.1 Hypotheses	398	(1)
14.2 The Martin Kernel and the Martin Space	399	(4)
14.3 Minimal Points and Boundary Limits	403	(1)
14.4 The Martin Representation	404	(4)
14.5 Applications	408	(2)
14.6 The Martin Boundary for Brownian Motion	410	(1)
14.7 The Dirichlet Problem in the Martin Space	411	(3)
Exercises	413	(1)
Notes on Chapter 14	414	(2)
Chapter 15 The Basis of Duality: A Fireside Chat	416	(5)
15.1 Duality Measures	416	(1)
15.2 The Cofine Topology	417	(3)
Notes on Chapter 15	420	(1)
Bibliography	421	(5)
Index	426