Logarithmic Sobolev Inequalities
After Poincaré inequalities, logarithmic Sobolev inequalities are amongst the most studied functional inequalities for semigroups. They contain much more information than Poincaré inequalities, and are at the same time sufficiently general to be available
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Logarithmic Sobolev Inequalities
After Poincaré inequalities, logarithmic Sobolev inequalities are amongst the most studied functional inequalities for semigroups. Indeed, they contain much more information than Poincaré inequalities, and are at the same time sufficiently general to be available in numerous cases of interest, in particular in infinite dimension (as limits of Sobolev inequalities on finite-dimensional spaces). Moreover, they entail remarkable smoothing properties of the semigroup in the form of hypercontractivity. The structure of this chapter is quite similar to the preceding one on Poincaré inequalities. In particular, the setting is that of a Full Markov Triple, abbreviated as “Markov Triple”, (E, μ, ) with associated Dirichlet form E, infinitesimal generator L, Markov semigroup P = (Pt )t≥0 and underlying function algebras A0 and A (cf. Sect. 3.4, p. 168). Logarithmic Sobolev inequalities under the invariant measure only concern finite (normalized) measures μ. It should be mentioned that logarithmic Sobolev inequalities involve entropy and Fisher information, which deal with strictly positive functions. It will therefore be convenient to deal with the class Aconst+ of Remark 3.3.3, p. 154, consisting of functions which are sums of a positive 0 function in A0 and a strictly positive constant. Once again, several definitions and properties make sense for more general triples (E, μ, ), and the Standard Markov Triple assumption suffices for a number of results. Contrary to the preceding chapter on Poincaré inequalities, the diffusion property can however not be discarded. The first section introduces the basic definition of a logarithmic Sobolev inequality together with its first properties. Section 5.2 presents the exponential decay in entropy and the fundamental equivalence between logarithmic Sobolev inequality and hypercontractivity. The next sections discuss integrability properties of eigenvectors and of Lipschitz functions under a logarithmic Sobolev inequality and present, as in the case of Poincaré inequalities, a criterion for measures on the real line to satisfy a logarithmic Sobolev inequality (for the usual gradient). Sections 5.5 and 5.7 deal with curvature conditions, first for the local logarithmic Sobolev inequalities for heat kernel measures, then for the invariant measure with additional dimensional information. Local hypercontractivity and some applications of the local logarithmic Sobolev inequalities towards heat kernel bounds are further presented. Section 5.6 D. Bakry et al., Analysis and Geometry of Markov Diffusion Operators, Grundlehren der mathematischen Wissenschaften 348, DOI 10.1007/978-3-319-00227-9_5, © Springer International Publishing Switzerland 2014
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Logarithmic Sobolev Inequalities
develops Harnack-type inequalities under the infinite-dimensional curvature conditions CD(ρ, ∞), and describes links with reverse local logarithmic Sobolev inequalities.
5.1 Logarithmic Sobolev Inequalities This section formally introduces the notion of a logarithmi
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