Preface

Contents

1 Preliminaries

1.1 Preliminaries in probability

1.1.1 Probability space

1.1.2 Random variable and probability distribution

1.1.3 Mathematical expectation and momentum

1.2 Some preliminaries of stochastic process

1.2.1 Markov process

1.2.2 Preliminaries on ergodic theory

1.3 Martingale

1.4 Wiener process and Brown motion

1.5 Poisson process

1.6 Lévy process

1.6.1 Characteristic function and infinite divisibility

1.6.2 Lévy process

1.6.3 Lévy–Itô decomposition

1.7 The fractional Brownian motion

2 The stochastic integral and Itô formula

2.1 Stochastic integral

2.1.1 Itô integral

2.1.2 The stochastic integral in general case

2.1.3 Poisson stochastic integral

2.2 Itô formula

2.3 The infinite-dimensional case

2.3.1 Q-Wiener process and the stochastic integral

2.3.2 Itô formula

2.4 Nuclear operator and HS operator

3 OU processes and SDEs

3.1 Ornstein–Uhlenbeck processes

3.2 Linear SDEs

3.3 Nonlinear SDEs

4 Random attractors

4.1 Determinate nonautonomous systems

4.2 Stochastic dynamical systems

5 Applications

5.1 Stochastic GL equation

5.1.1 The existence of random attractor

5.1.2 Hausdorff dimension of random attractor

5.1.3 Generalized SGLE

5.2 Ergodicity for SGL with degenerate noise

5.2.1 Momentum estimate and pathwise uniqueness

5.2.2 Invariant measures

5.2.3 Ergodicity

5.2.4 Some remarks

5.3 Stochastic damped forced Ostrovsky equation

5.3.1 Introduction

5.3.2 Well-posedness

5.3.3 Uniform estimates of solutions

5.3.4 Asymptotic compactness and random attractors

5.4 Simplified quasi-geostrophic model

5.4.1 The existence and uniqueness of solution

5.4.2 Existence of random attractors

5.5 Stochastic primitive equations

5.5.1 Stochastic 2D primitive equations with Lévy noise

5.5.2 Large deviation for stochastic primitive equations

Bibliography

Index