随机动力系统导论

Chapter 1 Introduction

1.1 Examples of deterministic dynamical systems

1.2 Examples of stochastic dynamical systems

1.3 Mathematical modeling with stochastic differential equations

1.4 Outline of this book

1.5 Problems

Chapter 2 Background in Analysis and Probability

2.1 Euclidean space

2.2 Hilbert, Banach and metric spaces

2.3 Taylor expansions

2.4 Improper integrals and Cauchy principal values

2.5 Some useful inequalities

2.5.1 Young's inequality

2.5.2 Cronwall inequality

2.5.3 Cauchy-Schwaxz inequality

2.5.4 HSlder inequality

2.5.5 Minkowski inequality

2.6 HSlder spaces, Sobolev spaces and related inequalities

2.7 Probability spaces

2.7.1 Scalar random variables

2.7.2 Random vectors

2.7.3 Gaussian random variables

2.7.4 Non-Gaussian random variables

2.8 Stochastic processes

2.9 Coovergence concepts

2.10 Simulation

2.11 Problems

Chapter 3 Noise

3.1 Brownian motion

3.1.1 Brownian motion in R1

3.1.2 Brownian motion in Rn~

3.2 What is Gaussian white noise

3.3* A mathematical model for Gaussian white noise

3.3.1 Generalized derivatives

3.3.2 Gaussian white noise

3.4 Simulation

3.5 Problems

Chapter 4 A Crash Course in Stochastic Differential Equations

4.1 Differential equations with noise

4.2 Riemann-Stieltjes integration

4.3 Stochastic integration and stochastic differential equations

4.3.1 Motivation

4.3.2 Definition of It5 integral

4.3.3 Practical calculations

4.3.4 Stratonovich integral

4.3.5 Examples

4.3.6 Properties of It6 integrals

4.3.7 Stochastic differential equations

4.3.8 SDEs in engineering and science literature

4.3.9 SDEs with two-sided Brownian motions

4.4 It's formula

4.4.1 Motivation for stochasticChain rules

4.4.2 ItS's formula in scalar case

4.4.3 It6's formula in vector case

4.4.4 Stochastic product rule and integration by parts

4.5 Linear stochastic differential equations

4.6 Nonlinear stochastic differential equations

4.6.1 Existence, uniqueness and smoothness

4.6.2 Probability measure px and expectation Ex associated with an SDE

4.7 Conversion between It5 and Stratonovich stochastic differential equations

4.7.1 Scalar SDEs

4.7.2 SDE systems

4.8 Impact of noise on dynamics

4.9 Simulation

4.10 Problems

Chapter 5 Deterministic Quantities for Stochastic Dynamics

5.1 Moments

5.2 Probability density functions

5.2.1 Scalar Fokker-Planck equations

5.2.2 Multidimensional Fokker-Planck equations

5.2.3 Existence and uniqueness for Fokker-Planck equations

5.2.4 Likelihood for transitions between different dynamical regimes under uncertainty

5.3 Most probable phase portraits

5.3.1 Mean phase portraits

5.3.2 Almost sure phase portraits

5.3.3 Most probable phase portraits

5.4 Mean exit time

5.5 Escape probability

5.6 Problems

Chapter 6 Invariant Structures for Stochastic Dynamics

6.1 Deterministic dynamical systems

6.1.1 Concepts for deterministic dynamical systems

6.1.2 The Haxtman-Grobman theorem

6.1.3 Invariant sets

6.1.4 Differentiable manifolds

6.1.5 Deterministic invariant manifolds

6.2 Measurable dynamical systems

6.3 Random dynamical systems

6.3.1 Canonical sample spaces for SDEs

6.3.2 Wiener shift

6.3.3 Cocycles and random dynamical systems

6.3.4 Examples of cocycles

6.3.5 Structural stability and stationary orbits

6.4 Linear stochastic dynamics

6.4.1 Oseledets' multiplicative ergodic theorem and Lyapunov exponents"

6.4.2 A stochastic Hartman-Grobman theorem

6.5* Random invariant manifolds

6.5.1 Definition of random invariant manifolds

6.5.2 Converting SDEs to RDEs

6.5.3 Local random pseudo-stable and pseudo-unstable manifolds

6.5.4 Local random stable, unstable and center manifolds

6.6 Problems

Chapter 7 Dynamical Systems Driven by Non-Gaussian Levy Motions

7.1 Modeling via stochastic differential equations with Levy motions

7.2 Levy motions

7.2.1 Functions that have one-side limits

7.2.2 Levy-Ito decomposition

7.2.3 Levy-Khintchine formula

7.2.4 Basic properties of Levy motions

7.3 s-stable Levy motions

7.3.1 Stable random variables

7.3.2 a-stable Levy motions in R1

7.3.3 a-stable Levy motion in Rn

7.4 Stochastic differential equations with Levy motions

7.4.1 Stochastic integration with respect to Levy motions

7.4.2 SDEs with Levy motions

7.4.3 Generators for SDEs with Levy motion

7.5 Mean exit time

7.5.1 Mean exit time for a-stable Levy motion

7.5.2 Mean exit time for SDEs with a-stable Levy motion

7.6 Escape probability and transition phenomena

7.6.1 Balayage-Dirichlet problem for escape probability

7.6.2 Escape probability for a-stable Levy motion

7.6.3 Escape probability for SDEs with a-stable Levy motion

7.7 Fokker-Planck equations

7.7.1 Fokker-Planck equations in R1

7.7.2 Fokker-Planck equations in Rn

7.8 Problems

Hints and Solutions

Further Readings

References

Index

Color Pictures

随机动力系统是一个入门较难的新兴领域。
　　《纯粹数学与应用数学专著：随机动力系统导论（英文）》是这个领域的一个较为通俗易懂的引论。
　　在《纯粹数学与应用数学专著：随机动力系统导论（英文）》的第一部分，作者从简单的随机动力系统实际例子出发，引导读者回顾概率论和白噪声的基本知识，深入浅出地介绍随机微积分，然后自然地展开随机微分方程的讨论。