UID:
almahu_9947366158402882
Format:
1 online resource (527 p.)
ISBN:
1-281-76312-8
,
9786611763121
,
0-08-087441-X
Series Statement:
Pure and applied mathematics ; 122
Content:
Nonstandard methods in stochastic analysis and mathematical physics
Note:
Description based upon print version of record.
,
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
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3.2. Hyperfinite Probability Spaces3.3. 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
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Chapter 5. Hyperfinite Dirichlet Forms and Markov Processes5.1. 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
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6.5. Applications of Nonstandard Analysis to the Boltzmann Equation6.6. 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
,
English
Additional Edition:
ISBN 0-12-048860-4
Language:
English
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