圖書信息
出版社: 科學出版社; 第1版 (2005年1月1日)
叢書名: 當代傑出青年科學文庫
精裝: 379頁
正文語種: 英語
開本: 16
ISBN: 7030144155
條形碼: 9787030144157
尺寸: 23.8 x 17.4 x 2.8 cm
重量: 721 g
作者簡介
王風雨，博士，1966年12月生於安徽省嘉山縣。北京師範大學教授，長江學者特聘教授，博士生導師，國家傑出青年科學基金獲得者。
曾應邀訪問英國Warwick大學，還應邀訪問過美國、法國、德國、俄羅斯、日本、新加坡、義大利和台灣等國家和地區的20餘所大學和研究所，並多次在國際學術會議上作邀請報告。目前，擔任中國機率統計學會常務理事，美國《數學評論》和德國《數學文摘》評論員，《套用機率統計》等雜誌的編委。曾經作為洪堡學者在德國Bidefeld大學工作。曾經獲得鍾嘉慶數學獎，教育部科技進步獎一等獎，國家自然科學三等獎和教育部首屆高校青年教師獎，霍英東青年教師獎研究類一等獎。獲得北京市五四青年獎章，人選首批新世紀百千萬工程國家級人才計畫，承擔國家重點基礎研究發展規劃“973”項目。研究方向涉及機率論、微分幾何、統計物理和泛函分析等多個學科領域，已發表論文近80篇，出版專著一部。
內容簡介
《泛函不等式馬爾可夫半群與譜理論》內容簡介：In this book, we introduce functional inequalities to describe:
（i） the spectrum of the generator: the essential and discrete spectrums,high order eigenvalues, the principal eigenvalue, and the spectral gap;
（ii） the semigroup properties: the uniform integrability, the compactness,the convergence rate, and the existence of density;
（iii） the reference measure and the intrinsic metric: the concentration, the isoperimetric inequality, and the transportation cost inequality.
目錄
Chapter 0 Preliminaries
0.1 Dirichlet forms, sub-Markov semigroups and generators
0.2 Dirichlet forms and Markov processes
0.3 Spectral theory
0.4 Riemannian geometry
Chapter 1 Poincaré Inequality and Spectral Gap
1.1 A general result and examples
1.2 Concentration of measures
1.3 Poincaré inequalities for jump processes
1.3.1 The bounded jump case
1.3.2 The unbounded jump case
1.3.3 A criterion for birth-death processes
1.4 Poincaré inequality for diffusion processes
1.4.1 The one-dimensional case
1.4.2 Spectral gap for diffusion processes on R上標d
1.4.3 Existence of the spectral gap on manifolds and application to nonsymmetric elliptic operators
1.5 Notes
Chapter 2 Diffusion Processes on Manifolds and Applications
2.1 kendall-Cranston's coupling
2.2 Estimates of the first （closed and Neumann） eigenvalue
2.3 Estimates of the first two Dirichlet eigenvalues
2.3.1 Estimates of the first Dirichlet eigenvalue
2.3.2 Estimates of the second Dirichlet eigenvalue and the spectralgap
2.4 Gradient estimates of diffusion semigroups
2.4.1 Gradient estimates of the closed and Neumann semigroups
2.4.2 Gradient estimates of Dirichlet semigroups
2.5 Harnack and isoperimetric inequalities using gradient estimates
2.5.1 Gradient estimates and the dimension-free Harnack inequality
2.5.2 The first eigenvalue and isoperimetric constants
2.6 Liouville theorems and couplings on manifolds
2.6.1 Liouville theorem using the Brownian radial process
2.6.2 Liouville theorem using the derivative formula
2.6.3 Liouville theorem using the conformal change of metric
2.6.4 Applications to harmonic maps and coupling Harmonic maps
2.7 Notes
Chapter 3 Functional Inequalities and Essential Spectrum
3.1 Essential spectrum on Hilbert spaces
3.1.1 Functional inequalities
3.1.2 Application to nonsymmetric semigroups
3.1.3 Asymptotic kernels for compact operators
3.1.4 Compact Markov operators without kernels
3.2 Applications to coercive closed forms
3.3 Super Poincaré inequalities
3.3.1 The F-Sobolev inequality
3.3.2 Estimates of semigroups
3.3.3 Estimates of high order eigenvalues
3.3.4 Concentration of measures for super Poincaré inequalities
3.4 criteria for super Poincaré inequalities
3.4.1 A localization method
3.4.2 Super Poincaré inequalities for jump processes
3.4.3 Estimates of β for diffusion processes
3.4.4 Some examples for estimates of high order eigenvalues
3.4.5 Some criteria for diffusion processes
3.5 Notes
Chapter 4 Weak Poicaré Inequalities and Convergence of Semigroups
4.1 General results
4.2 Concentration of measures
4.3 Criteria of weak Poincaré inequalities
4.4 Isoperimetric inequalities
4.4.1 Diffusion processes on manifolds
4.4.2 Jump processes
4.5 Notes
Chapter 5 Log-Sobolev Inequalities and Semigroup Properties
5.2 Spectral gap for hyperbounded operators
5.3 Concentration of measures for log-Sobolev inequalities
5.4 Logarithmic Sobolev inequalities for jump processes
5.4.1 Isoperimetric inequalities
5.4.2 Criteria for birth-death processes
5.5 Logarithmic Sobolev inequalities for one-dimensional diffusion processes
5.6 Estimates of the log-Sobolev constant on manifolds
5.6.1 Equivalent statements for the curvature condition
5.6.2 Estimates of α（V） using Bakry-Emery's criterion
5.6.3 Estimates of α（V） using Harnack inequality
5.6.4 Estimates of α（V） using coupling
5.7 Criteria of hypercontractivity, superboundedness and ultraboundedness
5.7.1 Some criteria
5.7.2 Ultraboundedness by perturbations
5.7.3 Isoperimetric inequalities
5.7.4 Some examples
5.8 Strong ergodicity and log-Sobolev inequality
5.9 Notes
Chapter 6 Interpolations of Poincaré and Log-Sobolev Inequalities
6.1 Some properties of （6.0.3）
6.2 Some criteria of （6.0.3）
6.3 Transportation cost inequalities
6.3.1 Otto-Villani's coupling
6.3.2 Transportation cost inequalities
6.3.3 Some results on （I下標p）
6.4 Notes
Chapter 7 Some Infinite Dimensional Models
7.1 The （weighted） Poisson spaces
7.1.1 Weak Poincaréinequalities for second Quantization Dirichlet forms
7.1.2 A class of jump processes on configuration spaces
7.1.3 Functional inequalities for ε上標Г下標J
7.2 Analysis on path spaces over Riemannian manifolds
7.2.1 Weak Poincaré inequality on finite-time interval path spaces
7.2.2 Weak Poincaré inequality on infinite-time interval path spaces
7.2.3 Transportation cost inequality on path spaces with L上標2-distance
7.2.4 Transportation cost inequality on path spaces with the intrinsic distance
7.3 Functional and Harnack inequalities for generalized Mehler semigroups
7.3.1 Some general results
7.3.2 Some examples
7.3.3 A generalized Mehler semigroup associated with the Dirichlet heat semigroup
7.4 Notes
Bibliography
Index
