**Mathematical Surveys and Monographs**

Volume: 64;
1999;
334 pp;
Hardcover

MSC: Primary 60;
Secondary 35

**Print ISBN: 978-0-8218-0806-1
Product Code: SURV/64**

List Price: $64.00

AMS Member Price: $51.20

MAA Member Price: $57.60

**Electronic ISBN: 978-1-4704-2853-2
Product Code: SURV/64.E**

List Price: $44.00

AMS Member Price: $35.20

MAA Member Price: $39.60

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# Stochastic Partial Differential Equations: Six Perspectives

Share this page *Edited by *
*Rene A. Carmona; Boris Rozovskii*

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.

#### Readership

Graduate students and researchers working in probability theory, PDEs, fluid dynamics, turbulence, chaos, particle systems, population biology, nonlinear filtering and financial mathematics.

#### Table of Contents

# Table of Contents

## Stochastic Partial Differential Equations: Six Perspectives

- Contents vii8
- Preface ix10
- PART 1. Stochastic Models 114
- Chapter 1. Stochastic Partial Differential Equations:Selected Applications in Continuum Physics 316
- Chapter 2. Measure-Valued Processes and Renormalization of Branching Particle Systems 4558
- Chapter 3. Deterministic and Stochastic Hydrodynamic Equations Arising From Simple Microscopic Model Systems 107120
- 1. Introduction 107120
- PART I: Non Reversible Dynamical Systems: Asymmetric Models with Shocks 111124
- PART II: Reversible Dynamical Systems: Symmetric Models with Long Range Interactions 128141
- Reversible Dynamical Systems: Symmetric Models with Long Range Interactions 128141
- 6. Ising Models with Kac Potentials: Glauber and Kawasaki Dynamics 128141
- 7. Nonlinear Fluctuations: Stochastic Allen-Cahn and Cahn-Hilliard Equations 135148
- 8. Macroscopic Effects of Small Fluctuations: the Origin of Spatial Patterns 141154
- 9. The Dynamics on Very Long Times: a Brief Look at Large Deviations 148161

- Bibliography 149162

- Chapter 4. Transport by Incompressible Random Velocity Fields: Simulations & Mathematical Conjectures 153166
- 1. Introduction 153166
- 2. Gaussian Velocity Fields with Kolmogorov Spectra 155168
- 3. Abstract Ornstein Uhlenbeck Velocity Fields 157170
- 4. Simulation of the Velocity Field 159172
- 5. Transport Simulations 170183
- 6. Homogenization & Spectral Singularity Renormalization 176189
- 7. Poisson Models 178191
- Bibliography 179192

- PART 2. Mathematical Theory 183196
- Chapter 5. An analytic approach to SPDE's 185198
- 1. Introduction 185198
- 2. Generalities 186199
- 3. The Stochastic Banach Spaces 190203
- 4. Model Equations 196209
- 5. Equations with Variable Coefficients 207220
- 6. Proof of Theorem 5.1 214227
- 7. Embedding Theorems for H[sup(n)][sub(p)](τ) 220233
- 8. Applications 225238
- 9. Open Problems 240253
- Bibliography 241254

- Chapter 6. Martingale Problems for Stochastic PDE's 243256

- Indexes 327340