Nonstandard methods in stochastic analysis and mathematical physics

목차 (Table of Contents)

• CONTENTS
• Preface = ⅸ
• Part I. BASIC COURSE
• Chapter 1. Calculus
• 1.1 Infinitesimals = 4
• 1.2 The Extended Universe = 15
• 1.3 Limits, Continuity, and the Derivative = 23
• 1.4 The Integral = 27
• 1.5 Differential Equations = 30
• References = 43
• Chapter 2. Topology and Linear Spaces
• 2.1 Topology and Saturation = 45
• 2.2 Linear Spaces and Operators = 53
• 2.3 Spectral Decomposition of Compact Hermitian Operators = 56
• 2.4 Nonstandard Methods in Banach Space Theory = 59
• References = 62
• Chapter 3. Probability
• 3.1 The Loeb Measure = 63
• 3.2 Hyperfinite Probability Spaces = 67
• 3.3 Brownian Motion = 78
• 3.4 Pushing Down Loeb Measures = 86
• 3.5 Applicaitons to Limit Measures and Measure Extensions = 95
• References = 103
• Part II. SELECTED APPLICATIONS
• Chapter 4. Stochastic Analysis
• 4.1 The Hyperfinite It$$\hat o$$Integral = 107
• A. Stochastic Integration = 108
• B. Liftings = 111
• C. It$$\hat o$$Lemma = 113
• 4.2 General Theory of Stochastic Integration = 115
• A. Internal Martingales = 115
• B. Martingale Integration = 122
• 4.3 Lifting Theorems = 130
• A. Nonanticipating Liftings = 130
• B. Uniform Liftings = 135
• 4.4 Representation Theorems = 141
• A. Square S-Integrable Martingales = 141
• B. Nonstandard Representations of Standard stochastic Integrals = 145
• C. Quadratic Variation and It$$\hat o$$Lemma = 148
• D. Further Remarks on Nonstandard Representations = 152
• F. Standard Representations of Nonstandard Stochastic Integrals = 153
• 4.5 Stochastic Differential Equations = 159
• A. It$$\hat o$$Equations with Continuous Coefficients = 159
• B. It$$\hat o$$Equations with Measurable Coefficients = 163
• C. Equations with Coefficients Depending on the Past = 170
• 4.6 Optimal Stochastic Controls = 171
• A. Optimal Controls : The Markov Case = 172
• B. Girsanov's Formula = 182
• C. Optimal Controls : Dependence on the Past = 189
• 4.7 Stochastic Integration in Infinite-Dimensional Spaces = 193
• A. Brownian Motion on Hilbert Spaces = 195
• B. Infinite-Dimensional Stochastic Integrals = 197
• C. A Remark on Stochastic Partial Differential Equations = 202
• 4.8 Whit Noise and L$$\acute e$$vy Brownian Motion = 205
• A. Construction of White Noise = 206
• B. Stochastic Integrals and the Continuity Theorem = 207
• C. L$$\acute e$$vy Brownian Motion = 213
• D. Invariance Principles = 216
• Notes = 220
• References = 220
• Chapter 5. Hyperfinite Dirichlet Forms and Markov Processes
• 5.1 Hyperfinite Quadratic Forms and Their Domains = 226
• A. The Domain = 226
• B. The Resolvent = 234
• 5.2 Connections to Standard Theory = 241
• 5.3 Hyperfinite Dirichlet Forms = 247
• A. Hyperfinite Markov Processes and the Definiton or Dirichlet Forms = 248
• B. Alternative Descriptions of Dirichlet Forms = 250
• C. Equilibrium Potentials = 255
• D. Fukushima's Decompostion Theorem = 258
• E. The Hyperfinite Feynman-Kac Formula = 260
• 5.4 Standard Parts and Markov Processes = 264
• A. Exceptional Sets = 265
• B. Strong Markov Processes and Modified Standard Parts = 273
• 5.5 Regular Forms and Markov Processes = 281
• A. Separaton of Compacts = 282
• B. Nearstandardly Concentrated Forms = 285
• C. Quasi-Coninuous Extensions = 287
• D. Regular Forms = 291
• 5.6 Applications to Quantum Mechanics and Stochastic Differential Equations = 297
• A. Hamiltonians and Energy Forms = 297
• B. Standard and Nonstandard Energy Forms = 302
• C. Energy Forms and Markov Processes = 310
• References = 315
• Chapter 6. Topics in Differential Operators
• 6.1 A Singular Sturm-Liouville Problem = 319
• 6.2 Singular Perturbations of Non-Negative Operators = 327
• A. The Computaton = 330
• B. Nontriviality = 333
• C. The Main Result = 336
• D. The Case of Standard λ's = 340
• E. Translation into Standard Terms = 343
• 6.3 Point Interactions = 349
• A. Application of the General Theory = 350
• B. An Alternative Approach = 352
• 6.4 Perturbations by Local Time Functionals = 358
• A. Applications of the General Theory = 359
• B. The Basic Estimate = 364
• C. Models of Polymers = 372
• 6.5 Applications of Nonstandard Analysis to the Boltzmann Equation = 379
• A. The Equation = 379
• B. Physically Natural Initial Conditions = 382
• C. The Equation with a Truncated Collision Term = 383
• D. The Nonstandard Tool = 385
• E. The Space-Homogeneous Case : Global Existence of Solutions = 386
• F. The Space-Homogeneous Case : Asymptotic Convergence to Equilibrium = 389
• G. The Space-Inhomogeneous Case : A Loeb-Measure Approach = 395
• 6.6 A Final Remark on the Feynman Path Integral and Other Matters = 403
• References = 407
• Chapter 7. Hyperfinite Lattice Models
• 7.1 Stochastic Evolution of Lattice Systems = 402
• 7.2 Equilibrium Theory = 419
• 7.3 The Global Markov Property = 431
• A. Hyperfinite Markov Property = 432
• B. Lifting and the Global Markov Property = 434
• C. S-Continuity, Dobrushin's Condition, and the Global Markov Property = 438
• D. Maximal and Minimal Gibbs States = 441
• E. The Case of Unbounded Fiber = 445
• 7.4 Hyperfinite Models for Quantum Field Theory = 449
• A. The Program = 449
• B. Free Scalar Fields = 453
• C. Interaction Scalar Fields = 458
• D. Some Concluding Remarks on Gauge Fields = 468
• 7.5 Fields and Polymers = 476
• A. Poisson Fields of Brownian Bridges = 477
• B. The Square of the Free Field as a Local Time Functional = 482
• C. Local Time Representations for Interactions Which Are Functions of $$Φ^2$$ = 486
• D. $$Φ^4$$and Polymer Measures = 493
• E. $$\mathop Φ_1^2$$$$\mathop Φ_2^2$$ = 496
• References = 504
• Index = 509