擴(kuò)散過(guò)程和偏微分方程

內(nèi)容概要

本書(shū)是根據(jù)作者在Sophia大學(xué)、Hokkaido大學(xué)、Tohoku大學(xué)、Tokyo Metropolitan大學(xué)和Tsukuba大學(xué)的講義整理而成，主旨是系統(tǒng)介紹概率論中馬爾可夫過(guò)程構(gòu)造問(wèn)題的現(xiàn)代分析方法，即將馬爾可夫過(guò)程的構(gòu)造歸結(jié)為研究二階退化橢圓微分方程的的邊值問(wèn)題。為此本書(shū)詳細(xì)介紹諸如：可測(cè)函數(shù)與函數(shù)空間、廣義函數(shù)、策分流形、擬微分算子、退化橢圓微分方程的極值原理、橢圓邊界問(wèn)題，F(xiàn)eller半群等現(xiàn)代分析中基本內(nèi)容與研究方法，最后介紹將這些理論與方法應(yīng)用于馬爾可夫過(guò)程構(gòu)造問(wèn)題。這是一部分析概率與偏微分程方面的優(yōu)秀專(zhuān)著。

書(shū)籍目錄

Preface Acknowledgments Notation and Conventions Iatroduetion and Summary Ⅰ Markov Processes and Semigroups Ⅱ Propagation of Maximums Ⅲ Construction of Feller Semigroups Chapter1 Preparatory Material   1.1 Sets   1.2 Mappings   1.3 Topological Spaces   1.4 Compactness   1.5 Connectedness   1.6 Metric Spaces   1.7 Baire‘s Category Theorem   1.8 Continuous Mappings   1.9 Linear Spaces   1.10 Linear Topological Spaces   1.11 Factor Spaces   1.12 Algebras and Modules   1.13 Linear Operators  1.14 Differentiable Mappings  1.15 Vector Fields and Integral Cuives  1.16 Measurable Spaces  1.17 Measurable Functions  1.18 Measures  1.19 Integrals  1.20 Probability Spaces  NotesChapter2 Manifolds,Tensors and Densicties  2.1 Manifolds  2.2 C Mappings  2.3 Tangent Bundles  2.4 Vector Fields  2.5 Integral Curves  2.6 Cotangent Bundles  2.7 Tensors  2.8 Tensor Fields  2.9 Exterior Product  2.10 Differential Forms  2.11 Densities  2.12 Integration on Manifolds  2.13 Manifolds With Boundary  NotesChapter3 Functional Analysis  3.1 Quasinormed Linear Spaces  3.2 Normed Linear Spaces  3.3 The Riesz Representation Theorem  3.4 Closed Operators  3.5 Complemented Subspaces  3.6 Compact Operators  3.7 Fredholm Operators  3.8 Hilbert Spaces  3.9 Theory 9of Semigroups  NotesChapter4 Distributions,Operators and Kernels  4.1 Notation  ……Chapter5 Sobolev SpacesChapter6 The Claculus of Pseudo-Differential OperatorsChapter7 Maximum Principles for Degenerate Elliptic OperatorsChapter8 Elliptic Boundary Value ProblemsChapter9 Markov Processes,Semigroups and Boundary Value ProblemsChapter10 Construction of Feller SemigroupsBibliographyList of SymbolsIndex

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