Nonstandard Methods in Stochastic Analysis and Mathematical Physics
Published on 28. July 1986
511 pages
978-0-08-087441-8 (ISBN)
Description
Nonstandard Methods in Stochastic Analysis and Mathematical Physics
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Sergio Albeverio
Nonstandard Methods in Stochastic Analysis and Mathematical Physics
Book
12/1986
Academic Press
€114.52
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Content
- Front Cover
- 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
