Previously issued in print: 2014.

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. 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.

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Half Title; Title Page; Copyright; Dedication; Contents; Preface; 1 Introduction; 1.1 Motivation; 1.2 Examples of Stochastic Partial Differential Equations; 1.3 Outlines for This Book; 1.3.1 Chapter 2: Deterministic Partial Differential Equations; 1.3.2 Chapter 3: Stochastic Calculus in Hilbert Space; 1.3.3 Chapter 4: Stochastic Partial Differential Equations; 1.3.4 Chapter 5: Stochastic Averaging Principles; 1.3.5 Chapter 6: Slow Manifold Reduction; 1.3.6 Chapter 7: Stochastic Homogenization; 2 Deterministic Partial Differential Equations; 2.1 Fourier Series in Hilbert Space.

2.2 Solving Linear Partial Differential Equations2.3 Integral Equalities; 2.4 Differential and Integral Inequalities; 2.5 Sobolev Inequalities; 2.6 Some Nonlinear Partial Differential Equations; 2.6.1 A Class of Parabolic PDEs; 2.6.1.1 Outline of the Proof of Theorem 2.4; 2.6.2 A Class of Hyperbolic PDEs; 2.6.2.1 Outline of the Proof of Theorem 2.5; 2.7 Problems; 3 Stochastic Calculus in Hilbert Space; 3.1 Brownian Motion and White Noise in Euclidean Space; 3.1.1 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 Space3.5 Brownian Motion and White Noise in Hilbert Space; 3.5.1 White Noise in Hilbert Space; 3.6 Stochastic Calculus in Hilbert Space; 3.7 It�o'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; 4.3.1 Mild Solutions of Nonautonomous spdes; 4.3.2 Mild Solutions of Autonomous spdes; 4.3.2.1 Formulation; 4.3.2.2 Well-Posedness Under Global Lipschitz Condition; 4.3.2.3 Well-Posedness Under Local Lipschitz Condition; 4.3.2.4 An Example.

4.4 Martingale Solutions4.5 Conversion Between It�o and Stratonovich SPDEs; 4.5.1 Case of Scalar Multiplicative Noise; 4.5.2 Case of General Multiplicative Noise; 4.5.3 Examples; 4.6 Linear Stochastic Partial Differential Equations; 4.6.1 Wave Equation with Additive Noise; 4.6.2 Heat Equation with Multiplicative Noise; 4.7 Effects of Noise on Solution Paths; 4.7.1 Stochastic Burgers' Equation; 4.7.2 Likelihood for Remaining Bounded; 4.8 Large Deviations for SPDEs; 4.9 Infinite Dimensional Stochastic Dynamics; 4.9.1 Basic Concepts; 4.9.2 More Dynamical Systems Concepts.

4.10 Random Dynamical Systems Defined by SPDEs4.10.1 Canonical Probability Space for SPDEs; 4.10.2 Perfection of Cocycles; 4.10.3 Examples; 4.11 Problems; 5 Stochastic Averaging Principles; 5.1 Classical Results on Averaging; 5.1.1 Averaging in Finite Dimension; 5.1.2 Averaging in Infinite Dimension; 5.2 An Averaging Principle for Slow-Fast SPDEs; 5.3 Proof of the Averaging Principle Theorem 5.20; 5.3.1 Some a priori Estimates; 5.3.2 Averaging as an Approximation; 5.4 A Normal Deviation Principle for Slow-Fast SPDEs; 5.5 Proof of the Normal Deviation Principle Theorem 5.34.