Franck Barthe

We develop the optimal transportation approach to modified log-Sobolev inequalities and to isoperimetric inequalities. Various sufficient conditions for such inequalities are given. Some of them are new even in the classical log-Sobolev case. The idea behind many of these conditions is that measures with a non-convex potential may enjoy such functional inequalities provided they have a strong integrability property that balances the lack of convexity. In addition, several known criteria are recovered in a simple unified way by transportation methods and generalized to the Riemannian setting.

We study in details the isoperimetric profile of product probability measures with tails between the exponential and the Gaussian regime. In particular we exhibit many examples where coordinate half-spaces are approximate solutions of the isoperimetric problem.

We introduce and study a notion of Orlicz hypercontractive semigroups. We analyze their relations with general F -Sobolev inequalities, thus extending Gross hypercontractivity theory. We provide criteria for these Sobolev type inequalities and for related properties. In particular, we implement in the context of probability measures the ideas of Maz'ja's capacity theory, and present equivalent forms relating the capacity of sets to their measure. Orlicz hypercontractivity efficiently describes the integrability improving properties of the Heat semigroup associated to the Boltzmann measures \mu_{\alpha}(dx) = (Z_{\alpha})^{-1} e^{-2|x|^{\alpha}} dx , when \alpha\in (1,2) . As an application we derive accurate isoperimetric inequalities for their products. This completes earlier works by Bobkov-Houdré and Talagrand, and provides a scale of dimension free isoperimetric inequalities as well as comparison theorems.

We provide a sufficient condition for a measure on the real line to satisfy a modified logarithmic Sobolev inequality, thus extending the criterion of Bobkov and G\"{o}tze. Under mild assumptions the condition is also necessary. Concentration inequalities are derived. This completes the picture given in recent contributions by Gentil, Guillin and Miclo.

It is shown that if X 1 , X 2 , … X_1,X_2,\ldots are independent and identically distributed square-integrable random variables, then the entropy of the normalized sum \[ Ent ⁡ ( X 1 + ⋯ + X n n ) \operatorname {Ent} \left (\frac {X_{1}+\cdots + X_{n}}{\sqrt {n}} \right ) \] is an increasing function of n n . The result also has a version for non-identically distributed random variables or random vectors.