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An Introduction to Stochastic Dynamics(随机动力系统导论)


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An Introduction to Stochastic Dynamics(随机动力系统导论)
  • 书号：9787030438577
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  • 装帧：平脊精装
    开本：B5
  • 页数：300
    字数：373
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    出版时间：null
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  • 定价： ￥128.00元
    售价： ￥101.12元
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目录

  • Contents
    Chapter 1Introduction1
    1.1Examples of deterministic dynamical systems1
    1.2Examples of stochastic dynamical systems8
    1.3Mathematical modeling with stochastic differential equations10
    1.4Outline of this book11
    1.5Problems12
    Chapter 2Background in Analysis and Probability13
    2.1Euclidean space13
    2.2Hilbert, Banach and metric spaces14
    2.3Taylor expansions15
    2.4Improper integrals and Cauchy principal values16
    2.5Some useful inequalities18
    2.5.1Young,s inequality18
    2.5.2Gronwall inequality18
    2.5.3Cauchy-Schwarz inequality19
    2.5.4Holder inequality20
    2.5.5Minkowski inequality 20
    2.6Holder spaces, Sobolev spaces and related inequalities21
    2.7Probability spaces25
    2.7.1Scalar random variables26
    2.7.2Random vectors27
    2.7.3Gaussian random variables29
    2.7.4Non-Gaussian random variables 30
    2.8Stochastic processes33
    2.9Convergence concepts35
    2.10Simulation36
    2.11 Problems38
    Chapter 3 Noise41
    3.1Brownian motion41
    3.1.1Brownian motion in R142
    3.1.2Brownian motion in Rn46
    3.2What is Gaussian white noise47
    3.3* A mathematical model for Gaussian white noise48
    3.3.1Generalized derivatives48
    3.3.2Gaussian white noise49
    3.4Simulation52
    3.5Problems 54
    Chapter 4A Crash Course in Stochastic Differential Equations57
    4.1Differential equations with noise57
    4.2Riemann-Stieltjes integration58
    4.3Stochastic integration and stochastic differential equations59
    4.3.1Motivation59
    4.3.2Definition of Ito integral61
    4.3.3Practical calculations62
    4.3.4Stratonovich integral63
    4.3.5Examples64
    4.3.6Properties of Ito integrals67
    4.3.7Stochastic differential equations69
    4.3.8SDEs in engineering and science literature70
    4.3.9SDEs with two-sided Brownian motions70
    4.4It6’sformula70
    4.4.1Motivation for stochastic chain rules70
    4.4.2It6’s formula in scalar case71
    4.4.3Ito’s formula in vector case74
    4.4.4Stochastic product rule and integration by parts76
    4.5Linear stochastic differential equations77
    4.6Nonlinear stochastic differential equations82
    4.6.1Existence, uniqueness and smoothness82
    4.6.2Probability measure and expectation Exassociated with an SDE 83
    4.7Conversion between Ito and Stratonovich stochastic differential equations84
    4.7.1Scalar SDEs84
    4.7.2SDE systems85
    4.8Impact of noise on dynamics86
    4.9Simulation88
    4.10Problems89
    Chapter 5Deterministic Quantities for Stochastic Dynamics93
    5.1Moments94
    5.2Probability density functions96
    5.2.1Scalar Fokker-Planck equations97
    5.2.2Multidimensional Fokker-Planck equations99
    5.2.3Existence and uniqueness for Fokker-Planck equations102
    5.2.4Likelihood for transitions between different dynamical regimes under uncertainty104
    5.3Most probable phase portraits104
    5.3.1Mean phase portraits104
    5.3.2Almost sure phase portraits105
    5.3.3Most probable phase portraits105
    5.4Mean exit time110
    5. 5Escape probability115
    5.6Problems121
    Chapter 6Invariant Structures for Stochastic Dynamics125
    6.1Deterministic dynamical systems126
    6.1.1Concepts for deterministic dynamical systems126
    6.1.2The Hartman-Grobman theorem128
    6.1.3Invariant sets128
    6.1.4Differentiable manifolds129
    6.1.5Deterministic invariant manifolds131
    6.2Measurable dynamical systems139
    6.3Random dynamical systems140
    6.3.1Canonical sample spaces for SDEs141
    6.3.2Wiener shift142
    6.3.3Cocycles and random dynamical systems143
    6.3.4Examples of cocycles146
    6.3.5Structural stability and stationary orbits148
    6.4Linear stochastic dynamics150
    6.4.1Oseledets’ multiplicative ergodic theorem and Lyapunov exponents150
    6.4.2A stochastic Hartman-Grobman theorem157
    6.5*Random invariant manifolds159
    6.5.1Definition of random invariant manifolds159
    6.5.2Converting SDEs to RDEs160
    6.5.3Local random pseudo-stable and pseudo-unstable manifolds163
    6.5.4Local random stable, unstable and center manifolds165
    6.6Problems168
    Chapter 7Dynamical Systems Driven by Non-Gaussian Levy Motions175
    7.1Modeling via stochastic differential equations withLevy motions176
    7.2Levy motions177
    7.2.1Functions that have one-side limits178
    7.2.2Levy-Ito decomposition179
    7.2.3Levy-Khintchine formula180
    7.2.4Basic properties of Levy motions181
    7.3a-stable Levy motions183
    7.3.1Stable random variables183
    7.3.2a-stable Levy motions in R1191
    7.3.3a-stable Levy motion in Rn197
    7.4Stochastic differential equations with Levy motions200
    7.4.1Stochastic integration with respect to Levy motions200
    7.4.2SDEs with Levy motions201
    7.4.3Generators for SDEs with Levy motion205
    7.5Mean exit time205
    7.5.1Mean exit time for a-stable Levy motion207
    7.5.2Mean exit time for SDEs with a-stable Levy motion210
    7.6Escape probability and transition phenomena213
    7.6.1Balayage-Dirichlet problem for escape probability213
    7.6.2Escape probability for a-stable Levy motion217
    7.6.3Escape probability for SDEs with a-stable Levy motion219
    7.7Fokker-Planck equations220
    7.7.1Fokker-Planck equations in R1221
    7.7.2Fokker-Planck equations in Rn223
    7.8Problems224
    Hints and Solutions228
    Further Readings255
    References257
    Index274
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