Nonstandard methods in stochastic analysis and mathematical physics, Volume 122 (Pure and Applied Mathematics)

✍ Scribed by Author Unknown (editor)

• Publisher: Academic Press
• Year: 1986
• Tongue: English
• Leaves: 527
• Edition: 1
• Category: Library

✦ Synopsis

The Bulletin of the American Mathematical Society acclaimed this text as "a welcome addition" to the literature of nonstandard analysis, a field related to number theory, algebra, and topology. The first half presents a self-contained introduction to the subject, and the second part explores applications to stochastic analysis and mathematical physics. 1986 edition.

✦ Table of Contents

Nonstandard Methods in Stochastic Analysis and Mathematical Physics
Copyright Page
Contents
Preface
Part I: Basic Course
Chapter 1. Calculus
1.1. Infinitesimals
1.2. The Extended Universe
1.3. Limits, Continuity, and the Derivative
1.4. The Integral
1.5. Differential Equations
References
Chapter 2. Topology and Linear Spaces
2.1. Topology and Saturation
2.2. Linear Spaces and Operators
2.3. Spectral Decomposition of Compact Hermitian Operators
2.4. Nonstandard Methods in Banach Space Theory
References
Chapter 3. Probability
3.1. The Loeb Measure
3.2. Hyperfinite Probability Spaces
3.3. Brownian Motion
3.4. Pushing Down Loeb Measures
3.5. Applications to Limit Measures and Measure Extensions
References
Part II: Selected Applications
Chapter 4. Stochastic Analysis
4.1. The Hyperfinite Itô Integral
4.2. General Theory of Stochastic Integration
4.3. Lifting Theorems
4.4. Representation Theorems
4.5. Stochastic Differential Equations
4.6. Optimal Stochastic Controls
4.7. Stochastic Integration in Infinite-Dimensional Spaces
4.8. White Noise and Lévy Brownian Motion
Notes
References
Chapter 5. Hyperfinite Dirichlet Forms and Markov Processes
5.1. Hyperfinite Quadratic Forms and Their Domains
5.2. Connections to Standard Theory
5.3. Hyperfinite Dirichlet Forms
5.4. Standard Parts and Markov Processes
5.5. Regular Forms and Markov Processes
5.6. Applications to Quantum Mechanics and Stochastic Differential Equations
References
Chapter 6. Topics in Differential Operators
6.1. A Singular Sturm–Liouville Problem
6.2. Singular Perturbations of Non-Negative Operators
6.3. Point Interactions
6.4. Perturbations by Local Time Functionals
6.5. Applications of Nonstandard Analysis to the Boltzmann Equation
6.6. A Final Remark on the Feynman Path Integral and Other Matters
References
Chapter 7. Hyperfinite Lattice Models
7.1. Stochastic Evolution of Lattice Systems
7.2. Equilibrium Theory
7.3. The Global Markov Property
7.4. Hyperfinite Models for Quantum Field Theory
7.5. Fields and Polymers
References
Index