Multi-Parameter Processes:
An Introduction to Random Fields
By Davar Khoshnevisan
Table of Contents:
Part I: Discrete-Parameter Random Fields
Chapter 1. Discrete Parameter Martingales
- One-Parameter Martingales
- Ortho-Martingales
- Martingales
- Supplementary Exercises
- Notes on Chapter 1
Chapter 2. Two Applications in Analysis
- Haar Systems
- Differentiation
- Supplementary Exercises
- Notes on Chapter 2
Chapter 3. Random Walks
- One-Parameter Random Walks
- Intersection Probabilities
- The Simple Random Walk
- Supplementary Exercises
- Notes on Chapter 3
Chapter 4. Multiparameter Random Walks
- The Strong Law of Large Numbers
- The Law of the Iterated Logarithm
- Supplementary Exercises
- Notes on Chapter 4
Chapter 5. Gaussian Random Variables
- The Basic Construction
- Regularity Theory
- Standard Brownian Sheet
- Supplementary Exercises
- Notes on Chapter 5
Chapter 6. Limit Theorems
- Random Variables
- Weak Convergence
- The Space C
- Invariance Principles
- Supplementary Exercises
- Notes on Chapter 6
Part II: Continuous-Parameter Random Fields
Chapter 7. Continuous Parameter Martingales
- One-Parameter Martingales
- Multiparameter Martingales
- One-Parameter Stochastic Integration
- Stochastic Partial Differential Equations
- Supplementary Exercises
- Notes on Chapter 7
Chapter 8. Constructing Markov Processs
- Discrete Markov Chains
- Markov Semigroups
- Markov Processes
- Feller Processes
- Supplementary Exercises
- Notes on Chapter 8
Chapter 9. Generation of Markov Processes
- Generation
- Explicit Computations
- The Feynman-Kac Formula
- Exit Times and Brownian Motion
- Supplementary Exercises
- Notes on Chapter 9
Chapter 10. Probabilistic Potential Theory
- Recurrent Lévy Processes
- Hitting Probabilities for Feller Processes
- Explicit Computations
- Supplementary Exercises
- Notes on Chapter 10
Chapter 11. Multiparameter Markov Processes
- Definitions
- Examples
- Potential Theory
- Applications
- Alpha-Regular Gaussian Random Fields
- Supplementary Exercises
- Notes on Chapter 11
Chapter 12. Brownian Sheet and Potential Theory
- Polar Sets of the Range of Brownian Sheet
- The Codimension of the Level Sets
- Local Times as Frostman's Measures
- Supplementary Exercises
- Notes on Chapter 12
Part III: Appendices
- Kolmogorov's Consistency Theorem
- Laplace Transforms
- Hausdorff Dimensions and Measures
- Energy and Capacity
Bibliography
References
Subject Index
Name Index