Preface ........................................................ ix
1 Introduction ................................................. 1
1.1 Motivation .............................................. 1
1.2 Examples of Stochastic Partial Differential Equations ... 2
1.3 Outlines for This Book .................................. 4
2 Deterministic Partial Differential Equations ................. 7
2.1 Fourier Series in Hilbert Space ......................... 7
2.2 Solving Linear Partial Differential Equations ........... 8
2.3 Integral Equalities .................................... 11
2.4 Differential and Integral Inequalities ................. 13
2.5 Sobolev Inequalities ................................... 13
2.6 Some Nonlinear Partial Differential Equations .......... 16
2.7 Problems ............................................... 19
3 Stochastic Calculus in Hilbert Space ........................ 21
3.1 Brownian Motion and White Noise in Euclidean Space ..... 21
3.2 Deterministic Calculus in Hilbert Space ................ 26
3.3 Random Variables in Hilbert Space ...................... 32
3.4 Gaussian Random Variables in Hilbert Space ............. 34
3.5 Brownian Motion and White Noise in Hilbert Space ....... 35
3.6 Stochastic Calculus in Hilbert Space ................... 39
3.7 Ito's Formula in Hilbert Space ......................... 42
3.8 Problems ............................................... 45
4 Stochastic Partial Differential Equations ................... 49
4.1 Basic Setup ............................................ 49
4.2 Strong and Weak Solutions .............................. 51
4.3 Mild Solutions ......................................... 52
4.4 Martingale Solutions ................................... 62
4.5 Conversion Between Ito and Stratonovich SPDEs .......... 65
4.6 Linear Stochastic Partial Differential Equations ....... 71
4.7 Effects of Noise on Solution Paths ..................... 77
4.8 Large Deviations for SPDEs ............................. 80
4.9 Infinite Dimensional Stochastic Dynamics ............... 82
4.10 Random Dynamical Systems Defined by SPDEs .............. 84
4.11 Problems ............................................... 88
5 Stochastic Averaging Principles ............................. 93
5.1 Classical Results on Averaging ......................... 93
5.2 An Averaging Principle for Slow-Fast SPDEs ............ 107
5.3 Proof of the Averaging Principle Theorem 5.20 ......... 112
5.4 A Normal Deviation Principle for Slow-Fast SPDEs ...... 118
5.5 Proof of the Normal Deviation Principle Theorem 5.34 .. 121
5.6 Macroscopic Reduction for Stochastic Systems .......... 128
5.7 Large Deviation Principles for the Averaging
Approximation ......................................... 132
5.8 PDEs with Random Coefficients ......................... 135
5.9 Further Remarks ....................................... 139
5.10 Looking Forward ....................................... 141
5.11 Problems .............................................. 141
6 Slow Manifold Reduction .................................... 145
6.1 Background ............................................ 145
6.2 Random Center-Unstable Manifolds for Stochastic
Systems ............................................... 150
6.3 Random Center-Unstable Manifold Reduction ............. 159
6.4 Local Random Invariant Manifold for SPDEs ............. 165
6.5 Random Slow Manifold Reduction for Slow-Fast SPDEs .... 170
6.6 A Different Reduction Method for SPDEs: Amplitude
Equation .............................................. 181
6.7 Looking Forward ....................................... 183
6.8 Problems .............................................. 184
7 Stochastic Homogenization .................................. 187
7.1 Deterministic Homogenization .......................... 188
7.2 Homogenized Macroscopic Dynamics for Stochastic
Linear Microscopic Systems ............................ 197
7.3 Homogenized Macroscopic Dynamics for Stochastic
Nonlinear Microscopic Systems ......................... 224
7.4 Looking Forward ....................................... 229
7.5 Problems .............................................. 229
Hints and Solutions ........................................... 233
Notations ..................................................... 255
References .................................................... 257
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