Functional inequalities, Markov semigroups and spectral theory

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Title:

Personal Author:

Wang, Feng-Yu

Publication Information:

Beijing : Science Press, 2005

ISBN:

9780080449425

Subject Term:

Inequalities (Mathematics)

Semigroups

Markov processes

Spectral theory (Mathematics)

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Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010103183	QA295 W36 2005	Open Access Book	Book	Searching... Unknown

In this book, the functional inequalities are introduced to describe:(i) the spectrum of the generator: the essential and discrete spectrums, high order eigenvalues, the principle eigenvalue, and the spectral gap;(ii) the semigroup properties: the uniform intergrability, the compactness, the convergence rate, and the existence of density;(iii) the reference measure and the intrinsic metric: the concentration, the isoperimetic inequality, and the transportation cost inequality.

Chapter 0 Preliminaries	p. 1
0.1 Dirichlet forms, sub-Markov semigroups and generators	p. 1
0.2 Dirichlet forms and Markov processes	p. 6
0.3 Spectral theory	p. 9
0.4 Riemannian geometry	p. 16
Chapter 1 Poincare Inequality and Spectral Gap	p. 24
1.1 A general result and examples	p. 24
1.2 Concentration of measures	p. 26
1.3 Poincare inequalities for jump processes	p. 31
1.3.1 The bounded jump case	p. 32
1.3.2 The unbounded jump case	p. 35
1.3.3 A criterion for birth-death processes	p. 43
1.4 Poincare inequality for diffusion processes	p. 45
1.4.1 The one-dimensional case	p. 45
1.4.2 Spectral gap for diffusion processes on R[superscript d]	p. 50
1.4.3 Existence of the spectral gap on manifolds and application to nonsymmetric elliptic operators	p. 56
1.5 Notes	p. 64
Chapter 2 Diffusion Processes on Manifolds and Applications	p. 67
2.1 Kendall-Cranston's coupling	p. 67
2.2 Estimates of the first (closed and Neumann) eigenvalue	p. 78
2.3 Estimates of the first two Dirichlet eigenvalues	p. 86
2.3.1 Estimates of the first Dirichlet eigenvalue	p. 86
2.3.2 Estimates of the second Dirichlet eigenvalue and the spectral gap	p. 88
2.4 Gradient estimates of diffusion semigroups	p. 93
2.4.1 Gradient estimates of the closed and Neumann semigroups	p. 93
2.4.2 Gradient estimates of Dirichlet semigroups	p. 97
2.5 Harnack and isoperimetric inequalities using gradient estimates	p. 108
2.5.1 Gradient estimates and the dimension-free Harnack inequality	p. 108
2.5.2 The first eigenvalue and isoperimetric constants	p. 111
2.6 Liouville theorems and couplings on manifolds	p. 114
2.6.1 Liouville theorem using the Brownian radial process	p. 114
2.6.2 Liouville theorem using the derivative formula	p. 116
2.6.3 Liouville theorem using the conformal change of metric	p. 120
2.6.4 Applications to harmonic maps and coupling Harmonic maps	p. 121
2.7 Notes	p. 123
Chapter 3 Functional Inequalities and Essential Spectrum	p. 127
3.1 Essential spectrum on Hilbert spaces	p. 127
3.1.1 Functional inequalities	p. 127
3.1.2 Application to nonsymmetric semigroups	p. 133
3.1.3 Asymptotic kernels for compact operators	p. 136
3.1.4 Compact Markov operators without kernels	p. 138
3.2 Applications to coercive closed forms	p. 142
3.3 Super Poincare inequalities	p. 145
3.3.1 The F-Sobolev inequality	p. 145
3.3.2 Estimates of semigroups	p. 150
3.3.3 Estimates of high order eigenvalues	p. 158
3.3.4 Concentration of measures for super Poincare inequalities	p. 160
3.4 Criteria for super Poincare inequalities	p. 163
3.4.1 A localization method	p. 163
3.4.2 Super Poincare inequalities for jump processes	p. 165
3.4.3 Estimates of [beta] for diffusion processes	p. 168
3.4.4 Some examples for estimates of high order eigenvalues	p. 173
3.4.5 Some criteria for diffusion processes	p. 178
3.5 Notes	p. 181
Chapter 4 Weak Poincare Inequalities and Convergence of Semigroups	p. 182
4.1 General results	p. 182
4.2 Concentration of measures	p. 189
4.3 Criteria of weak Poincare inequalities	p. 193
4.4 Isoperimetric inequalities	p. 199
4.4.1 Diffusion processes on manifolds	p. 199
4.4.2 Jump processes	p. 203
4.5 Notes	p. 206
Chapter 5 Log-Sobolev Inequalities and Semigroup Properties	p. 208
5.1 Three boundedness properties of semigroups	p. 208
5.2 Spectral gap for hyperbounded operators	p. 215
5.3 Concentration of measures for log-Sobolev inequalities	p. 225
5.4 Logarithmic Sobolev inequalities for jump processes	p. 229
5.4.1 Isoperimetric inequalities	p. 229
5.4.2 Criteria for birth-death processes	p. 231
5.5 Logarithmic Sobolev inequalities for one-dimensional diffusion processes	p. 234
5.6 Estimates of the log-Sobolev constant on manifolds	p. 236
5.6.1 Equivalent statements for the curvature condition	p. 236
5.6.2 Estimates of [alpha](V) using Bakry-Emery's criterion	p. 241
5.6.3 Estimates of [alpha](V) using Harnack inequality	p. 244
5.6.4 Estimates of [alpha](V) using coupling	p. 250
5.7 Criteria of hypercontractivity, superboundedness and ultraboundedness	p. 252
5.7.1 Some criteria	p. 252
5.7.2 Ultraboundedness by perturbations	p. 262
5.7.3 Isoperimetric inequalities	p. 267
5.7.4 Some examples	p. 270
5.8 Strong ergodicity and log-Sobolev inequality	p. 271
5.9 Notes	p. 276
Chapter 6 Interpolations of Poincare and Log-Sobolev Inequalities	p. 279
6.1 Some properties of (6.0.3)	p. 280
6.2 Some criteria of (6.0.3)	p. 285
6.3 Transportation cost inequalities	p. 291
6.3.1 Otto-Villani's coupling	p. 293
6.3.2 Transportation cost inequalities	p. 295
6.3.3 Some results on (I[subscript p])	p. 300
6.4 Notes	p. 304
Chapter 7 Some Infinite Dimensional Models	p. 306
7.1 The (weighted) Poisson spaces	p. 306
7.1.1 Weak Poincare inequalities for second quantization Dirichlet forms	p. 306
7.1.2 A class of jump processes on configuration spaces	p. 309
7.1.3 Functional inequalities for [epsilon superscript Gamma subscript J]	p. 314
7.2 Analysis on path spaces over Riemannian manifolds	p. 317
7.2.1 Weak Poincare inequality on finite-time interval path spaces	p. 317
7.2.2 Weak Poincare inequality on infinite-time interval path spaces	p. 327
7.2.3 Transportation cost inequality on path spaces with L[superscript 2]-distance	p. 331
7.2.4 Transportation cost inequality on path spaces with the intrinsic distance	p. 339
7.3 Functional and Harnack inequalities for generalized Mehler semigroups	p. 341
7.3.1 Some general results	p. 343
7.3.2 Some examples	p. 355
7.3.3 A generalized Mehler semigroup associated with the Dirichlet heat semigroup	p. 361
7.4 Notes	p. 362
Bibliography	p. 366
Index	p. 376

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