معادلات دیفرانسیل جزئی تصادفی: شش دیدگاه ; [کارگاه معادلات دیفرانسیل جزئی تصادفی که در دانشگاه کالیفرنیای جنوبی، لس آنجلس، در ژانویه 1996 برگزار شد]

The field of Stochastic Partial Differential Equations (SPDEs) is one of the most dynamically developing areas of mathematics. It lies at the cross section of probability, partial differential equations, population biology, and mathematical physics. The field is especially attractive because of its interdisciplinary nature and the enormous richness of current and potential future applications. This volume is a collection of six important topics in SPDEs presented from the viewpoint of distinguished scientists working in the field and related areas. Emphasized are the genesis and applications of SPDEs as well as mathematical theory and numerical methods. Contents 8 Preface 10 PART 1. Stochastic Models 14 Chapter 1. Stochastic Partial Differential Equations:Selected Applications in Continuum Physics 16 1. The Physical Basis of Stochastic Analysis 16 2. Mathematical and Computational Tools for Stochastic Analysis 26 3. Multi-phase Flow 34 4. Transport and Dispersion 47 Bibliography 53 Chapter 2. Measure-Valued Processes and Renormalization of Branching Particle Systems 58 1. Branching and Interacting Particle Systems 58 2. Historical Brownian Motion 71 3. Formulation of a General Class of Measure-Valued Branching Processes 74 4. Small Scale Behavior 77 5. Large Scale Behavior 87 6. A Survey of Interactive Branching Systems 97 Bibliography 115 Chapter 3. Deterministic and Stochastic Hydrodynamic Equations Arising From Simple Microscopic Model Systems 120 1. Introduction 120 PART I: Non Reversible Dynamical Systems: Asymmetric Models with Shocks 124 2. The Burgers Equation 124 3. The Asymmetric Simple Exclusion and the Independent Particle System 126 4. The Weakly Asymmetric Simple Exclusion Process: Hydrodynamics and Stochastic Corrections 132 5. Driven Surfaces and Fluctuations 139 PART II: Reversible Dynamical Systems: Symmetric Models with Long Range Interactions 141 Reversible Dynamical Systems: Symmetric Models with Long Range Interactions 141 6. Ising Models with Kac Potentials: Glauber and Kawasaki Dynamics 141 7. Nonlinear Fluctuations: Stochastic Allen-Cahn and Cahn-Hilliard Equations 148 8. Macroscopic Effects of Small Fluctuations: the Origin of Spatial Patterns 154 9. The Dynamics on Very Long Times: a Brief Look at Large Deviations 161 Bibliography 162 Chapter 4. Transport by Incompressible Random Velocity Fields: Simulations & Mathematical Conjectures 166 1. Introduction 166 2. Gaussian Velocity Fields with Kolmogorov Spectra 168 3. Abstract Ornstein Uhlenbeck Velocity Fields 170 4. Simulation of the Velocity Field 172 5. Transport Simulations 183 6. Homogenization & Spectral Singularity Renormalization 189 7. Poisson Models 191 Bibliography 192 PART 2. Mathematical Theory 196 Chapter 5. An analytic approach to SPDE's 198 1. Introduction 198 2. Generalities 199 3. The Stochastic Banach Spaces 203 4. Model Equations 209 5. Equations with Variable Coefficients 220 6. Proof of Theorem 5.1 227 7. Embedding Theorems for H[sup(n)][sub(p)](τ) 233 8. Applications 238 9. Open Problems 253 Bibliography 254 Chapter 6. Martingale Problems for Stochastic PDE's 256 1. Introduction 256 2. Stochastic Integrals for Cylindrical Martingales in Topological Vector Spaces 259 3. Martingale Problems 274 4. Equations of Stochastic Quantization 308 5. Appendix 330 Bibliography 336 Indexes 340 A 344 B 344 C 344 D 344 E 344 F 344 G 345 H 345 I 345 K 345 L 345 M 345 N 346 O 346 P 346 R 346 S 346 T 347 V 347 W 347 Notation Index 342 Subject Index 344 The field of Stochastic Partial Differential Equations (SPDEs) is one of the dynamically developing areas of mathematics. It lies at the cross section of probability, partial differential equations, population biology, and mathematical physics. This title emphasizes the genesis and applications of SPDEs as well as mathematical theory. The field of Stochastic Partial Differential Equations (SPDEs) is one of the most dynamically developing areas of mathematics. This volume offers a collection of six important topics in SPDEs. Emphasized are the genesis and applications of SPDEs as well as mathematical theory and numerical methods. Rene A. Carmona, Boris Rozovskii, Editors. Includes Bibliographical References And Indexes.