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。

Preface

1 Preliminaries, notations and conventions

1.1 Elements of topology

1.2 Measure theory

1.3 Functions of bounded variation. Riemann-Stieltjes integral

1.4 Sequences of independent random variables

1.5 Convex functions. Holder and Minkowski inequalities

1.6 The Cauchy equation

2 Basic notions in functional analysis

2.1 Linear spaces

2.2 Banach spaces

2.3 The space of bounded linear operators

3 Conditional expectation

3.1 Projections in Hilbert spaces

3.2 Definition and existence of conditional expectation

3.3 Properties and examples

3.4 The Radon-Nikodym Theorem

3.5 Examples of discrete martingales

3.6 Convergence of self-adjoint operators

3.7 ... and of martingales

4 Brownian motion and l-Iilbert spaces

4.1 Gaussian families & the definition of Brownian motion

4.2 Complete orthonormal sequences in a Hilbert space

4.3 Construction and basic properties of Brownian motion

4.4 Stochastic integrals

5 Dual spaces and convergence of probability measures

5.1 The Hahn-Banach Theorem

5.2 Form of linear functionals in specific Banach spaces

5.3 Thedual of an operator

5.4 Weak and weak* topologies

5.5 The Central Limit Theorem

5.6 Weak convergence in metric spaces

5.7 Compactness everywhere

5.8 Notes on other modes of convergence

6 The Gelfand transform and its applications

6.1 Banach algebras

6.2 The Gelfand transform

6.3 Examples of Gelfand transform

6.4 Examples of explicit calculations of Gelfand transform

6.5 Dense subalgebras of C(S)

6.6 Inverting the abstract Fourier transform

6.7 The Factorization Theorem

7 Semigroups of operators and Levy processes

7.1 The Banach-Steinhaus Theorem

7.2 Calculus of Banach space valued functions

7.3 Closed operators

7.4 Semigroups of operators

7.5 Brownian motion and Poisson process semigroups

7.6 More convolution semigroups

7.7 The telegraph process semigroup

7.8 Convolution semigroups of measures on semigroups

8 Markov processes and semigroups of operators

8.1 Semigroups of operators related to Markov processes

8.2 The Hille-Yosida Theorem

8.3 Generators of stochastic processes

8.4 Approximation theorems

9 Appendixes

9.1 Bibliographical notes

9.2 Solutions and hints to exercises

9.3 Some commonly used notations

References

Index