泛函不等式，马尔可夫半群与谱理论（英文版）

　　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上标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-Cranstons 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-Emerys 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-Villanis coupling
　　6.3.2 Transportation cost inequalities
　　6.3.3 Some results on (I下标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上标Г下标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上标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

　　    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。

　　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.