纯粹数学与应用数学专著：随机动力系统导论（英文）

图书标准信息

• 作者 段金桥 著
• 出版社 科学出版社
• 出版时间 2015-01
• 版次 1
• ISBN 9787030438577
• 定价 128.00元
• 装帧 平装
• 开本 16开
• 纸张 胶版纸
• 页数 283页
• 正文语种 英语
• 丛书 纯粹数学与应用数学专著

【内容简介】

　　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。

【目录】

Chapter1Introduction
1.1Examplesofdeterministicdynamicalsystems
1.2Examplesofstochasticdynamicalsystems
1.3Mathematicalmodelingwithstochasticdifferentialequations
1.4Outlineofthisbook
1.5Problems

Chapter2BackgroundinAnalysisandProbability
2.1Euclideanspace
2.2Hilbert,Banachandmetricspaces
2.3Taylorexpansions
2.4ImproperintegralsandCauchyprincipalvalues
2.5Someusefulinequalities
2.5.1Young'sinequality
2.5.2Cronwallinequality
2.5.3Cauchy-Schwaxzinequality
2.5.4HSlderinequality
2.5.5Minkowskiinequality
2.6HSlderspaces,Sobolevspacesandrelatedinequalities
2.7Probabilityspaces
2.7.1Scalarrandomvariables
2.7.2Randomvectors
2.7.3Gaussianrandomvariables
2.7.4Non-Gaussianrandomvariables
2.8Stochasticprocesses
2.9Coovergenceconcepts
2.10Simulation
2.11Problems

Chapter3Noise
3.1Brownianmotion
3.1.1BrownianmotioninR1
3.1.2BrownianmotioninRn~
3.2WhatisGaussianwhitenoise
3.3*AmathematicalmodelforGaussianwhitenoise
3.3.1Generalizedderivatives
3.3.2Gaussianwhitenoise
3.4Simulation
3.5Problems

Chapter4ACrashCourseinStochasticDifferentialEquations
4.1Differentialequationswithnoise
4.2Riemann-Stieltjesintegration
4.3Stochasticintegrationandstochasticdifferentialequations
4.3.1Motivation
4.3.2DefinitionofIt5integral
4.3.3Practicalcalculations
4.3.4Stratonovichintegral
4.3.5Examples
4.3.6PropertiesofIt6integrals
4.3.7Stochasticdifferentialequations
4.3.8SDEsinengineeringandscienceliterature
4.3.9SDEswithtwo-sidedBrownianmotions
4.4It'sformula
4.4.1MotivationforstochasticChainrules
4.4.2ItS'sformulainscalarcase
4.4.3It6'sformulainvectorcase
4.4.4Stochasticproductruleandintegrationbyparts
4.5Linearstochasticdifferentialequations
4.6Nonlinearstochasticdifferentialequations
4.6.1Existence,uniquenessandsmoothness
4.6.2ProbabilitymeasurepxandexpectationExassociatedwithanSDE
4.7ConversionbetweenIt5andStratonovichstochasticdifferentialequations
4.7.1ScalarSDEs
4.7.2SDEsystems
4.8Impactofnoiseondynamics
4.9Simulation
4.10Problems

Chapter5DeterministicQuantitiesforStochasticDynamics
5.1Moments
5.2Probabilitydensityfunctions
5.2.1ScalarFokker-Planckequations
5.2.2MultidimensionalFokker-Planckequations
5.2.3ExistenceanduniquenessforFokker-Planckequations
5.2.4Likelihoodfortransitionsbetweendifferentdynamicalregimesunderuncertainty
5.3Mostprobablephaseportraits
5.3.1Meanphaseportraits
5.3.2Almostsurephaseportraits
5.3.3Mostprobablephaseportraits
5.4Meanexittime
5.5Escapeprobability
5.6Problems

Chapter6InvariantStructuresforStochasticDynamics
6.1Deterministicdynamicalsystems
6.1.1Conceptsfordeterministicdynamicalsystems
6.1.2TheHaxtman-Grobmantheorem
6.1.3Invariantsets
6.1.4Differentiablemanifolds
6.1.5Deterministicinvariantmanifolds
6.2Measurabledynamicalsystems
6.3Randomdynamicalsystems
6.3.1CanonicalsamplespacesforSDEs
6.3.2Wienershift
6.3.3Cocyclesandrandomdynamicalsystems
6.3.4Examplesofcocycles
6.3.5Structuralstabilityandstationaryorbits
6.4Linearstochasticdynamics
6.4.1Oseledets'multiplicativeergodictheoremandLyapunovexponents"
6.4.2AstochasticHartman-Grobmantheorem
6.5*Randominvariantmanifolds
6.5.1Definitionofrandominvariantmanifolds
6.5.2ConvertingSDEstoRDEs
6.5.3Localrandompseudo-stableandpseudo-unstablemanifolds
6.5.4Localrandomstable,unstableandcentermanifolds
6.6Problems

Chapter7DynamicalSystemsDrivenbyNon-GaussianLevyMotions
7.1ModelingviastochasticdifferentialequationswithLevymotions
7.2Levymotions
7.2.1Functionsthathaveone-sidelimits
7.2.2Levy-Itodecomposition
7.2.3Levy-Khintchineformula
7.2.4BasicpropertiesofLevymotions
7.3s-stableLevymotions
7.3.1Stablerandomvariables
7.3.2a-stableLevymotionsinR1
7.3.3a-stableLevymotioninRn
7.4StochasticdifferentialequationswithLevymotions
7.4.1StochasticintegrationwithrespecttoLevymotions
7.4.2SDEswithLevymotions
7.4.3GeneratorsforSDEswithLevymotion
7.5Meanexittime
7.5.1Meanexittimefora-stableLevymotion
7.5.2MeanexittimeforSDEswitha-stableLevymotion
7.6Escapeprobabilityandtransitionphenomena
7.6.1Balayage-Dirichletproblemforescapeprobability
7.6.2Escapeprobabilityfora-stableLevymotion
7.6.3EscapeprobabilityforSDEswitha-stableLevymotion
7.7Fokker-Planckequations
7.7.1Fokker-PlanckequationsinR1
7.7.2Fokker-PlanckequationsinRn
7.8Problems

HintsandSolutions
FurtherReadings
References
Index
ColorPictures