泛函不等式,马尔可夫半群与谱理论

出版时间：2005-1  出版社：科学出版社  作者：王凤雨  页数：392  
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内容概要

本书的主要内容涉及概率论、泛函分析、微分几何和统计物理等多个学科，较系统的介绍了近十年有关泛函不等式及其近十年来的有关泛函不等式及其应用的主要研究成果和研究方法。其中的一些成果和研究思想被国际同行专家大量引用，引发了一系列的后续工作。以泛函不等式为主要工具研究马氏半群及其生成元的分析与概率性质。特别的，使用弱不等式刻画马氏半群的各种收敛速度；使用超不等式刻画半群一致可积性以及生成元的本征谱和高阶特征值的估计；引入一般型可加性的不等式，刻画马氏半群在不同意义下的指数式收敛以及概论距离的上界估计。

作者简介

　　王风雨，博士，1966年12月生于安徽省嘉山县。北京师范大学教授，长江学者特聘教授，博士生导师，国家杰出青年科学基金获得者。　　曾应邀访问英国Warwick大学，还应邀访问过美国、法国、德国、俄罗斯、日本、新加坡、意大利和台湾等国家和地区的20余所大学和研究所，并多次在国际学术会议上作邀请报告。目前，担任中国概率统计学会常务理事，美国《数学评论》和德国《数学文摘》评论员，《应用概率统计》等杂志的编委。曾经作为洪堡学者在德国Bidefeld大学工作。曾经获得钟嘉庆数学奖，教育部科技进步奖一等奖，国家自然科学三等奖和教育部首届高校青年教师奖，霍英东青年教师奖研究类一等奖。获得北京市五四青年奖章，人选首批新世纪百千万工程国家级人才计划，承担国家重点基础研究发展规划“973”项目。研究方向涉及概率论、微分几何、统计物理和泛函分析等多个学科领域，已发表论文近80篇，出版专著一部。

书籍目录

Chapter 0 Preliminaries0.1 Dirichlet forms, sub-Markov semigroups and generators0.2 Dirichlet forms and Markov processes0.3 Spectral theory0.4 Riemannian geometryChapter 1 Poincaré Inequality and Spectral Gap1.1 A general result and examples1.2 Concentration of measures1.3 Poincaré inequalities for jump processes1.3.1 The bounded jump case1.3.2 The unbounded jump case1.3.3 A criterion for birth-death processes1.4 Poincaré inequality for diffusion processes1.4.1 The one-dimensional case1.4.2 Spectral gap for diffusion processes on R上标d1.4.3 Existence of the spectral gap on manifolds and application to nonsymmetric elliptic operators1.5 NotesChapter 2 Diffusion Processes on Manifolds and Applications2.1 Kendall-Cranstons coupling2.2 Estimates of the first (closed and Neumann) eigenvalue2.3 Estimates of the first two Dirichlet eigenvalues2.3.1 Estimates of the first Dirichlet eigenvalue2.3.2 Estimates of the second Dirichlet eigenvalue and the spectralgap2.4 Gradient estimates of diffusion semigroups2.4.1 Gradient estimates of the closed and Neumann semigroups2.4.2 Gradient estimates of Dirichlet semigroups2.5 Harnack and isoperimetric inequalities using gradient estimates2.5.1 Gradient estimates and the dimension-free Harnack inequality2.5.2 The first eigenvalue and isoperimetric constants2.6 Liouville theorems and couplings on manifolds2.6.1 Liouville theorem using the Brownian radial process2.6.2 Liouville theorem using the derivative formula2.6.3 Liouville theorem using the conformal change of metric2.6.4 Applications to harmonic maps and coupling Harmonic maps2.7 NotesChapter 3 Functional Inequalities and Essential Spectrum3.1 Essential spectrum on Hilbert spaces3.1.1 Functional inequalities3.1.2 Application to nonsymmetric semigroups3.1.3 Asymptotic kernels for compact operators3.1.4 Compact Markov operators without kernels3.2 Applications to coercive closed forms3.3 Super Poincaré inequalities3.3.1 The F-Sobolev inequality3.3.2 Estimates of semigroups3.3.3 Estimates of high order eigenvalues3.3.4 Concentration of measures for super Poincaré inequalities3.4 Criteria for super Poincaré inequalities3.4.1 A localization method3.4.2 Super Poincaré inequalities for jump processes3.4.3 Estimates of β for diffusion processes3.4.4 Some examples for estimates of high order eigenvalues3.4.5 Some criteria for diffusion processes3.5 NotesChapter 4 Weak Poicaré Inequalities and Convergence of Semigroups4.1 General results4.2 Concentration of measures4.3 Criteria of weak Poincaré inequalities4.4 Isoperimetric inequalities4.4.1 Diffusion processes on manifolds4.4.2 Jump processes4.5 NotesChapter 5 Log-Sobolev Inequalities and Semigroup Properties5.2 Spectral gap for hyperbounded operators5.3 Concentration of measures for log-Sobolev inequalities5.4 Logarithmic Sobolev inequalities for jump processes5.4.1 Isoperimetric inequalities5.4.2 Criteria for birth-death processes5.5 Logarithmic Sobolev inequalities for one-dimensional diffusion processes5.6 Estimates of the log-Sobolev constant on manifolds5.6.1 Equivalent statements for the curvature condition5.6.2 Estimates of α(V) using Bakry-Emerys criterion5.6.3 Estimates of α(V) using Harnack inequality5.6.4 Estimates of α(V) using coupling5.7 Criteria of hypercontractivity, superboundedness and ultraboundedness5.7.1 Some criteria5.7.2 Ultraboundedness by perturbations5.7.3 Isoperimetric inequalities5.7.4 Some examples5.8 Strong ergodicity and log-Sobolev inequality5.9 NotesChapter 6 Interpolations of Poincaré and Log-Sobolev Inequalities6.1 Some properties of (6.0.3)6.2 Some criteria of (6.0.3)6.3 Transportation cost inequalities6.3.1 Otto-Villanis coupling6.3.2 Transportation cost inequalities6.3.3 Some results on (I下标p)6.4 NotesChapter 7 Some Infinite Dimensional Models7.1 The (weighted) Poisson spaces7.1.1 Weak Poincaréinequalities for second quantization Dirichlet forms7.1.2 A class of jump processes on configuration spaces7.1.3 Functional inequalities for ε上标Г下标J7.2 Analysis on path spaces over Riemannian manifolds7.2.1 Weak Poincaré inequality on finite-time interval path spaces7.2.2 Weak Poincaré inequality on infinite-time interval path spaces7.2.3 Transportation cost inequality on path spaces with L上标2-distance7.2.4 Transportation cost inequality on path spaces with the intrinsic distance7.3 Functional and Harnack inequalities for generalized Mehler semigroups7.3.1 Some general results7.3.2 Some examples7.3.3 A generalized Mehler semigroup associated with the Dirichlet heat semigroup7.4 NotesBibliographyIndex

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　　概率论与函数研究领域的研究生、教师和科研工作者。