Language: English
ISBN/ISSN: 9787510061462
2016-06; Soft Cover
Preface to the New Edition
Preface to the First Edition
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
Chapter2
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
Chapter4
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 Feynrnan-Kac Functional and the Schrodinger 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
Notes on Chapter 7
Chapter 8
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 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
Chapter9
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
Notes 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. t2 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
Preface to the New Edition
Preface to the First Edition
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
Chapter2
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
Chapter4
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 Feynrnan-Kac Functional and the Schrodinger 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
Notes on Chapter 7
Chapter 8
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 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
Chapter9
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
Notes 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. t2 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
