随机偏微分方程的有效动力学：英文

Preface1  Introduction1.1  Motivation1.2  Examples of Stochastic Partial Differential Equations1.3  Outlines for This Book2  Deterministic Partial Differential Equations2.1  Fourier Series in Hilbert Space2.2  Solving Linear Partial Differential Equations2.3  Integral Equalities2.4  Differential and Integral Inequalities2.5  Sobolev Inequalities2.6  Some Nonlinear Partial Differential Equations2.7  Problems3  Stochastic Calculus in Hilbert Space3.1  Brownian Motion and White Noise in Euclidean Space3.2  Deterministic Calculus in Hilbert Space3.3  Random Variables in Hilbert Space3.4  Gaussian Random Variables in Hilbert Space3.5  Brownian Motion and White Noise in Hilbert Space3.6  Stochastic Calculus in Hilbert Space3.7  It6's Formula in Hilbert Space3.8  Problems4  Stochastic Partial Differential Equations4.1  Basic Setup4.2  Strong and Weak Solutions4.3  Mild Solutions4.4  Martingale Solutions4.5  Conversion Between It6 and Stratonovich SPDEs4.6  Linear Stochastic Partial Differential Equations4.7  Effects of Noise on Solution Paths4.8  Large Deviations for SPDEs4.9  Infinite Dimensional Stochastic Dynamics4.10  Random Dynamical Systems Defined by SPDEs4.11  Problems5  Stochastic Averaging Principles5.1  Classical Results on Averaging5.2  An Averaging Principle for Slow-Fast SPDEs5.3  Proof of the Averaging Principle Theorem 5.205.4  A Normal Deviation Principle for Slow-Fast SPDEs5.5  Proof of the Normal Deviation Principle Theorem 5.345.6  Macroscopic Reduction for Stochastic Systems5.7  Large Deviation Principles for the Averaging Approximation5.8  PDEs with Random Coefficients5.9  Further Remarks5.10  Looking Forward5.11  Problems6  Slow Manifold Reduction6.1  Background6.2  Random Center-Unstable Manifolds for Stochastic Systems6.3  Random Center-Unstable Manifold Reduction6.4  Local Random Invariant Manifold for SPDEs6.5  Random Slow Manifold Reduction for Slow-Fast SPDEs6.6  A Different Reduction Method for SPDEs: Amplitude Equation6.7  Looking Forward6.8  Problems7  Stochastic Homogenization7.1  Deterministic Homogenization7.2  Homogenized Macroscopic Dynamics for Stochastic Linear Microscopic Systems7.3  Homogenized Macroscopic Dynamics for Stochastic Nonlinear Microscopic Systems7.4  Looking Forward7.5  ProblemsHints and SolutionsNotationsReferences