Functional Inequalities Markov Semigroups and Spectral Theory

Chapter 0: Preliminaries

Chapter 1 Poincáre Inequality and Spectral Gap

Chapter 2 Diffusion Processes on Manifolds and Applications

Chapter 3 Functional Inequalities and Essential Spectrum

Chapter 4: Weak Poincare Inequalities and Convergence of Semigroups
Chapter 5 Log-Sobolev Inequalities and Semigroup Properties

Chapter 6 Interpolations of Poincare and Log-Sobolev Inequalities

Chapter 7 Some Infinite Dimensional Models

Bibliography

Index

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.

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