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# An Introduction to Stochastic Dynamics

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Author: Jinqiao-Duan
Language: Chinese
ISBN/ISSN: 9787030438577
Published on: 2015-04
Hardcover

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 Gronwall inequality
2.5.3 Cauchy—Schwarz inequality
2.5.4 Holder inequality
2.5.5 Minkowski inequality
2.6 Holder 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 Convergence 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 Ito integral
4.3.3 Practical calculations
4.3.4 Stratonovich integral
4.3.5 Examples
4.3.6 Properties of Ito 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 Ito's formula
4.4.1 Motivation for stochastic chain rules
4.4.2 Ito's formula in scalar case
4.4.3 Ito'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 Probabilitv measure Px and expectation Ex associated with an SDE
4.7 Conversion between Ito 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 Hartman—Grobman theorem
6.1.3 Invariant sets
6.1.4 Differentiable manifolds
6.1.5 Deterministic invariant manifolds
6.2 Measurable dvnamical svstems
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 α—stable Levy motions
7.3.1 Stable random variables
7.3.2 α—stable Levy motions in R1
7.3.3 α—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 α—stable Levy motion
7.5.2 Mean exit time for SDEs with α—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 α—stable Levy motion
7.6.3 Escape probability for SDEs with α—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 