Effective Dynamics of Stochastic Partial Differential Equations - 1st Edition - ISBN: 9780128008829, 9780128012697

Effective Dynamics of Stochastic Partial Differential Equations

1st Edition

Authors: Jinqiao Duan Wei Wang
eBook ISBN: 9780128012697
Hardcover ISBN: 9780128008829
Imprint: Elsevier
Published Date: 27th February 2014
Page Count: 282
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Description

Effective Dynamics of Stochastic Partial Differential Equations focuses on stochastic partial differential equations with slow and fast time scales, or large and small spatial scales. The authors have developed basic techniques, such as averaging, slow manifolds, and homogenization, to extract effective dynamics from these stochastic partial differential equations.

The authors’ experience both as researchers and teachers enable them to convert current research on extracting effective dynamics of stochastic partial differential equations into concise and comprehensive chapters. The book helps readers by providing an accessible introduction to probability tools in Hilbert space and basics of stochastic partial differential equations. Each chapter also includes exercises and problems to enhance comprehension.

Key Features

  • New techniques for extracting effective dynamics of infinite dimensional dynamical systems under uncertainty
  • Accessible introduction to probability tools in Hilbert space and basics of stochastic partial differential equations
  • Solutions or hints to all Exercises

Readership

Students in applied mathematics and professionals in the sciences and engineering community need to deal with stochastic models

Table of Contents

Cover image

Dedication

Preface

1: Introduction

1.1 Motivation

1.2 Examples of Stochastic Partial Differential Equations

1.3 Outlines for This Book

2: Deterministic Partial Differential Equations

2.1 Fourier Series in Hilbert Space

2.2 Solving Linear Partial Differential Equations

2.3 Integral Equalities

2.4 Differential and Integral Inequalities

2.5 Sobolev Inequalities

2.6 Some Nonlinear Partial Differential Equations

2.7 Problems

3: Stochastic Calculus in Hilbert Space

3.1 Brownian Motion and White Noise in Euclidean Space

3.2 Deterministic Calculus in Hilbert Space

3.3 Random Variables in Hilbert Space

3.4 Gaussian Random Variables in Hilbert Space

3.5 Brownian Motion and White Noise in Hilbert Space

3.6 Stochastic Calculus in Hilbert Space

3.7 Itô’s Formula in Hilbert Space

3.8 Problems

4: Stochastic Partial Differential Equations

4.1 Basic Setup

4.2 Strong and Weak Solutions

4.3 Mild Solutions

4.4 Martingale Solutions

4.5 Conversion Between Itô and Stratonovich SPDEs

4.6 Linear Stochastic Partial Differential Equations

4.7 Effects of Noise on Solution Paths

4.8 Large Deviations for SPDEs

4.9 Infinite Dimensional Stochastic Dynamics

4.10 Random Dynamical Systems Defined by SPDEs

4.11 Problems

5: Stochastic Averaging Principles

5.1 Classical Results on Averaging

5.2 An Averaging Principle for Slow-Fast SPDEs

5.3 Proof of the Averaging Principle

5.4 A Normal Deviation Principle for Slow-Fast SPDEs

5.5 Proof of the Normal Deviation Principle

5.6 Macroscopic Reduction for Stochastic Systems

5.7 Large Deviation Principles for the Averaging Approximation

5.8 PDEs with Random Coefficients

5.9 Further Remarks

5.10 Looking Forward

5.11 Problems

6: Slow Manifold Reduction

6.1 Background

6.2 Random Center-Unstable Manifolds for Stochastic Systems

6.3 Random Center-Unstable Manifold Reduction

6.4 Local Random Invariant Manifold for SPDEs

6.5 Random Slow Manifold Reduction for Slow-Fast SPDEs

6.6 A Different Reduction Method for SPDEs: Amplitude Equation

6.7 Looking Forward

6.8 Problems

7: Stochastic Homogenization

7.1 Deterministic Homogenization

7.2 Homogenized Macroscopic Dynamics for Stochastic Linear Microscopic Systems

7.3 Homogenized Macroscopic Dynamics for Stochastic Nonlinear Microscopic Systems

7.4 Looking Forward

7.5 Problems

Hints and Solutions

Notations

References

Details

No. of pages:
282
Language:
English
Copyright:
© Elsevier 2014
Published:
Imprint:
Elsevier
eBook ISBN:
9780128012697
Hardcover ISBN:
9780128008829

About the Author

Jinqiao Duan

Illinois Institute of Technology, Chicago IL

Affiliations and Expertise

Illinois Institute of Technology, Chicago IL

Wei Wang

Affiliations and Expertise

Nanjing University, Nanjing, PR China.

Reviews

"Graduate students as well as researchers who have a solid background in SPDEs and are interested in undertaking research in the direction of averaging or homogenization of SPDEs can start their adventure and journey from the book…"--MathSciNet, Effective Dynamics of Stochastic Partial Differential Equations