Nonstandard Methods in Stochastic Analysis and Mathematical Physics (eBook)

Front Cover 1
Nonstandard Methods in Stochastic Analysis and Mathematical Physics 4
Copyright Page 5
Contents 6
Preface 10
Part I: Basic Course 14
Chapter 1. Calculus 16
1.1. Infinitesimals 17
1.2. The Extended Universe 28
1.3. Limits, Continuity, and the Derivative 36
1.4. The Integral 40
1.5. Differential Equations 43
References 56
Chapter 2. Topology and Linear Spaces 58
2.1. Topology and Saturation 58
2.2. Linear Spaces and Operators 66
2.3. Spectral Decomposition of Compact Hermitian Operators 69
2.4. Nonstandard Methods in Banach Space Theory 72
References 75
Chapter 3. Probability 76
3.1. The Loeb Measure 76
3.2. Hyperfinite Probability Spaces 80
3.3. Brownian Motion 91
3.4. Pushing Down Loeb Measures 99
3.5. Applications to Limit Measures and Measure Extensions 108
References 116
Part II: Selected Applications 118
Chapter 4. Stochastic Analysis 120
4.1. The Hyperfinite Itô Integral 120
4.2. General Theory of Stochastic Integration 128
4.3. Lifting Theorems 143
4.4. Representation Theorems 154
4.5. Stochastic Differential Equations 172
4.6. Optimal Stochastic Controls 184
4.7. Stochastic Integration in Infinite-Dimensional Spaces 206
4.8. White Noise and Lévy Brownian Motion 218
Notes 233
References 233
Chapter 5. Hyperfinite Dirichlet Forms and Markov Processes 238
5.1. Hyperfinite Quadratic Forms and Their Domains 239
5.2. Connections to Standard Theory 254
5.3. Hyperfinite Dirichlet Forms 260
5.4. Standard Parts and Markov Processes 277
5.5. Regular Forms and Markov Processes 294
5.6. Applications to Quantum Mechanics and Stochastic Differential Equations 310
References 328
Chapter 6. Topics in Differential Operators 332
6.1. A Singular Sturm–Liouville Problem 332
6.2. Singular Perturbations of Non-Negative Operators 340
6.3. Point Interactions 362
6.4. Perturbations by Local Time Functionals 371
6.5. Applications of Nonstandard Analysis to the Boltzmann Equation 392
6.6. A Final Remark on the Feynman Path Integral and Other Matters 416
References 420
Chapter 7. Hyperfinite Lattice Models 425
7.1. Stochastic Evolution of Lattice Systems 425
7.2. Equilibrium Theory 432
7.3. The Global Markov Property 444
7.4. Hyperfinite Models for Quantum Field Theory 462
7.5. Fields and Polymers 489
References 517
Index 522