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