经典位势论及其对应的概率论

出版时间：2013-6  出版社：世界图书出版公司北京公司  作者：Joseph L. Doob  

内容概要

本书是一部杰作，书中的第一部分重点讲述了势能理论及其相关的拉普拉斯方程和热方程；第二部分深入分析了和第一部分相关的随机过程理论部分。这部科学巨著，调理清楚、深刻明晰地研究了问题涉及的这两个方面，进而取代了零散地，语言上不一致的大量散落在各个图书馆的大量的书籍和科技文献，但并不是一个百科全书。书中并不是将所有的知识点简单相加，而是用自己独特的方式从基础开始进行系统讲述，所以学习这本书并不需要太多的基础准备。这是之前别的书不能企及的。有一位读者的如是评价：Doob是20世纪在概率论上做出开创性工作的大家，对随机过程的研究做出了杰出的贡献，尤其是对鞅的研究有深远影响。1950年代Wiley出版了Doob的第一本专著《随机过程》，总结了迄今为止的所有的重要工作，自此，随机过程研究界的一本“圣经”诞生了，前几年Wiley有再版了该书，平装本。
目次：（一）经典和抛物势能理论：经典势能理论的数学背景导引；调和、子调和和上和函数的基本形式；上和函数族的下确界；特殊开集上的势能；极集及其应用；基本收敛定理和约分；格林函数；相关调和函数的狄利克莱问题；格点和相关的函数类；扫除运作；.细拓扑；Martin边界；经典能和容量；一维势能；双曲势能；Slab上的子双曲、上双曲和双曲函数；双曲势能；双曲狄利克莱问题，清除和特殊集；双曲背景中的martin边界；（二）第一部分的概率对应部分：概率基础；任选次数和辅助概念；鞅理论基础；连续参数上鞅的基本性质；格点及其相关随机过程类；马尔科夫过程；布朗运动；Itô互积分；布朗运动和鞅理论；条件布朗运动；（三）经典势能理论和鞅理论中的格点；布朗运动和PWB方法；布朗运动Martin空间；附录：容量问题；格点问题；测度论中的格点理论概念；一致可积；核和过渡函数；积分极限定理；下半连续函数。
读者对象：物理、力学、数学以及概率统计专业的所有读者。

作者简介

作者:(美)杜布

书籍目录

Introduction Notation and Conventions Part 1 Classical and Parabolic Potential Theory Chapter I Introduction to the Mathematical Background of Classical Potential Theory 1.The Context of Green's Identity 2.Function Averages 3.Harmonic Functions  4.Maximum-Minimum Theorem for Harmonic Functions 5.The Fundamental Kernel for RN and Its Potentials 6.Gauss Integral Theorem 7.The Smoothness of Potentials ; The Poisson Equation 8.Harmonic Measure and the Riesz Decomposition Chapter II Basic Properties of Harmonic, Subharmonic, and Superharmonic Functions 1.The Green Function of a Ball; The Poisson Integral 2.Hamack's Inequality 3.Convergence of Directed Sets of Harmonic Functions 4.Harmonic, Subharmonic, and Superharmoruc Functions 5.Minimum Theorem for Superharmonic Functions 6.Application of the Operation TB 7.Characterization of Superharmonic Functions in Terms of Harmonic Functions 8.Differentiable Superharmonic Functions 9.Application of Jensen's Inequality 10.Superharmonic Funaions on an Annulus II.Examples 12.The Kelvin Transformation 13.Greenian Sets 14.The L1（uB_） and D（uB_） Classes of Harmonic Functions on a Ball B; The Riesz-Herglotz Theorem 15.The Fatou Boundary Limit Theorem 16.Minimal Harmonic Functions Chapter III Infima of Families of Superharmonic Functidns 1.Least Superharmonic Majorant （LM） and Greatest Subharmonic Minorant （GM）  2.Generalization of Theorem I 3.Fundamental Convergence Theorem （Preliminary Version） 4.The Reduction Operation 5.Reduction Properties 6.A Smallness Property of Reductions on Compact Sets 7.The Natural （Pointwise） Order Decomposition for Positive Superharmonk Functions  Chapter 1V Potentials on Special Open Sets 1.Special Open Sets, and Potentials on Them 2.Examples  3.A Fundamental Smallness Property of Potentials  4.Increasing Sequences of Potentials 5.Smoothing of a Potential 6.Uniqueness of the Measure Determining a Potential 7.Riesz Measure Associated with a Superharmonic Function 8.Riesz Decomposition Theorem 9.Counterpart for Superharmonic Functions on R2 ofthe Riesz Decomposition 10.An Approximation Theorem Chapter V Polar Sets and Their Applications 1.Definition 2.Superharmonic Functions Associated with a Polar Set 3.Countable Unions of Polar Sets 4.Properties ofPolar Sets 5.Extension of a Superharmonic Function 6.Greenian Sets in IR2 as the Complements of Nonpolar Sets 7.Superharmonic Function Minimum Theorem （Extension of Theorem I1.5） 8.Evans-Vasilesco Theorem 9.Approximation of a Potential by Continuous Potentials 10.The Domination Principle 11.The Infinity Set of a Potential and the Riesz Measure …… Part 2 Probabilistic Countrepart of Part 1 Part 3

章节摘录

版权页：   Filtered measurable spaces and their adapted families of functions provide a mathematical formalism modeling certain physical ideas. A measur.able space is a mathematical model of the set of possible events insome physical context, together with a distinguished class of compoundevents. If I is a subset of R, a filtration of is a mathe. matical model for the flow of events in time. Each pair represents apossible outcome of an experiment at time t, and (t) represents the classof compound events observable before or at time t. The value x of afunction x(t, .) at models the value of some observable at the outcome(t,ω), and the function x(t, .) itself is therefore incorporated in (t) insense that this function is supposed (t) measurable; that is, {x(.),(.)}is an adapted process. The Measurable Sets of a Topological Measurable SpaceIf a measurable space is given as a topological space, the a algebra of measur.able sets will always be the algebra of Borel subsets of the space unlesssome other algebra is specified. In particular, the state space R meansthe measurable space.

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《经典位势论及其对应的概率论(英文)》由世界图书出版公司北京公司出版。