Contents 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-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
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