概率论与随机过程中的泛函分析

Preface

1Preliminaries,notationsandconventions

1.1Elementsoftopology

1.2Measuretheory

1.3Functionsofboundedvariation.Riemann-Stieltjesintegral

1.4Sequencesofindependentrandomvariables

1.5Convexfunctions.HolderandMinkowskiinequalities

1.6TheCauchyequation

2Basicnotionsinfunctionalanalysis

2.1Linearspaces

2.2Banachspaces

2.3Thespaceofboundedlinearoperators

3Conditionalexpectation

3.1ProjectionsinHilbertspaces

3.2Definitionandexistenceofconditionalexpectation

3.3Propertiesandexamples

3.4TheRadon-NikodymTheorem

3.5Examplesofdiscretemartingales

3.6Convergenceofself-adjointoperators

3.7...andofmartingales

4Brownianmotionandl-Iilbertspaces

4.1Gaussianfamilies&thedefinitionofBrownianmotion

4.2CompleteorthonormalsequencesinaHilbertspace

4.3ConstructionandbasicpropertiesofBrownianmotion

4.4Stochasticintegrals

5Dualspacesandconvergenceofprobabilitymeasures

5.1TheHahn-BanachTheorem

5.2FormoflinearfunctionalsinspecificBanachspaces

5.3Thedualofanoperator

5.4Weakandweak*topologies

5.5TheCentralLimitTheorem

5.6Weakconvergenceinmetricspaces

5.7Compactnesseverywhere

5.8Notesonothermodesofconvergence

6TheGelfandtransformanditsapplications

6.1Banachalgebras

6.2TheGelfandtransform

6.3ExamplesofGelfandtransform

6.4ExamplesofexplicitcalculationsofGelfandtransform

6.5DensesubalgebrasofC(S)

6.6InvertingtheabstractFouriertransform

6.7TheFactorizationTheorem

7SemigroupsofoperatorsandLevyprocesses

7.1TheBanach-SteinhausTheorem

7.2CalculusofBanachspacevaluedfunctions

7.3Closedoperators

7.4Semigroupsofoperators

7.5BrownianmotionandPoissonprocesssemigroups

7.6Moreconvolutionsemigroups

7.7Thetelegraphprocesssemigroup

7.8Convolutionsemigroupsofmeasuresonsemigroups

8Markovprocessesandsemigroupsofoperators

8.1SemigroupsofoperatorsrelatedtoMarkovprocesses

8.2TheHille-YosidaTheorem

8.3Generatorsofstochasticprocesses

8.4Approximationtheorems

9Appendixes

9.1Bibliographicalnotes

9.2Solutionsandhintstoexercises

9.3Somecommonlyusednotations

References

Index

　　本书在介绍了泛函分析的基本概念后，用Hibert空间泛函的F.Riesz表示定理建立Radon-Nikodym定理，从而引进条件期望的概念；在Hilbert空间的正交分解概念的基础上，引进了Brown运动，并建立了随机积分的概念；证明了Hahn-Banach定理并引进了对偶空间的概念后，便可讨论概率分布的弱收敛及弱拓扑的紧性；在介绍了交换Banach代数的Gelfand表示后，讨论了抽象Fourier变换的反演公式。本书最后两章讨论了算子半群和Levy过程。证明了算子半群的Hille-Yosida定理，讨论了Markov过程与算子半群生成子的关系。　　本书是作者在Rice大学和Houston大学给研究生授课的讲义基础上写成的。本书在介绍了泛函分析的基本概念(如Banach空间)后，用Hibert空间泛函的F.Riesz表示定理建立Radon-Nikodym定理，从而引进条件期望的概念；在Hilbert空间的正交分解概念的基础上，引进了Brown运动，并建立了随机积分的概念；证明了Hahn-Banach定理并引进了对偶空间的概念后，便可讨论概率分布的弱收敛及弱拓扑的紧性；在介绍了交换Banach代数的Gelfand表示后，讨论了抽象Fourier变换的反演公式。本书最后两章讨论了算子半群和Levy过程。证明了算子半群的Hille-Yosida定理，讨论了Markov过程与算子半群生成子的关系。　　本书可作为高等学校本科高年级和研究生课程教材，对于专攻概率论与泛函分析的读者具有很好的参考价值，也可作为学习概率论和随机过程专著的导引。