STOCHASTIC PROCESSES AND APPLICATIONS-Pavliotis

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收藏 2023-12-21

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[size=14.666666984558105px]这是随机过程方面的著名教授Pavliotis的课堂笔记。内容涵盖了随机过程，扩散过程，随机微分方程，[size=14.666666984558105px]Fokker-Planck方程。讲解深入浅出，篇幅精炼，是随机微积分上手的好教材。

1 Stochastic Processes 1
1.1 DefinitionofaStochasticProcess ............................... 1 1.2 StationaryProcesses ...................................... 3 1.3 BrownianMotion........................................ 10 1.4 ExamplesofStochasticProcesses ............................... 13 1.5 TheKarhunen-Loe ́veExpansion ................................ 16 1.6 DiscussionandBibliography .................................. 20 1.7 Exercises ............................................ 21
2 Diffusion Processes 27
2.1 ExamplesofMarkovprocesses................................. 27 2.2 TheChapman-KolmogorovEquation.............................. 30 2.3 TheGeneratorofaMarkovProcesses ............................. 34 2.4 ErgodicMarkovprocesses ................................... 37 2.5 TheKolmogorovEquations................................... 39 2.6 DiscussionandBibliography .................................. 46 2.7 Exercises ............................................ 47
3 Introduction to SDEs 49
3.1 Introduction........................................... 49 3.2 TheItoˆandStratonovichStochasticIntegrals ......................... 52 3.3 SolutionsofSDEs........................................ 57 3.4 Itoˆ’sformula........................................... 59 3.5 ExamplesofSDEs ....................................... 65 3.6 Lamperti’sTransformationandGirsanov’sTheorem. . . . . . . . . . . . . . . . . . . . . . 68 3.7 LinearStochasticDifferentialEquations ............................ 70 3.8 DiscussionandBibliography .................................. 73 3.9 Exercises ............................................ 74
4 The Fokker-Planck Equation 77
4.1 BasicpropertiesoftheFokker-Planckequation ........................ 77 4.2 ExamplesoftheFokker-PlanckEquation ........................... 81
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4.3 DiffusionProcessesinOneDimension............................. 87 4.4 TheOrnstein-UhlenbeckProcess................................ 90 4.5 TheSmoluchowskiEquation .................................. 96 4.6 ReversibleDiffusions......................................101 4.7 EigenfunctionExpansions ...................................106 4.8 MarkovChainMonteCarlo...................................108 4.9 ReductiontoaSchro ̈dingerOperator..............................110 4.10 Discussion and Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.11Exercises ............................................118
Appendix A Frequently Used Notation 123
Appendix B Elements of Probability Theory 125

B.1 BasicDefinitionsfromProbabilityTheory...........................125
B.2 RandomVariables........................................127
B.3 ConditionalExpecation.....................................131
B.4 TheCharacteristicFunction...................................132
B.5 GaussianRandomVariables ..................................133
B.5.1 GaussianMeasuresinHilbertSpaces .........................135
B.6 TypesofConvergenceandLimitTheorems ..........................138
B.7 DiscussionandBibliography ..................................139