Logarithmic Sobolev inequality for diffusion semigroups

Through the main example of the Ornstein-Uhlenbeck semigroup, the Bakry-Emery criterion is presented as a main tool to get functional inequalities as Poincar\'e or logarithmic Sobolev inequalities. Moreover an alternative method using the optimal mass transportation, is also given to obtain the logarithmic Sobolev inequality

From the Pr\'ekopa-Leindler inequality to modified logarithmic Sobolev inequality

We develop in this paper an improvement of the method given by S. Bobkov and M. Ledoux. Using the Pr\'ekopa-Leindler inequality, we prove a modified logarithmic Sobolev inequality adapted for all measures on \dR^n, with a strictly convex and super-linear potential. This inequality implies modified logarithmic Sobolev inequality for all uniform strictly convex potential as well as the Euclidean logarithmic Sobolev inequality

Phi-entropy inequalities and Fokker-Planck equations

We present new $\Phi$-entropy inequalities for diffusion semigroups under the curvature-dimension criterion. They include the isoperimetric function of the Gaussian measure. Applications to the long time behaviour of solutions to Fokker-Planck equations are given

Phi-entropy inequalities for diffusion semigroups

We obtain and study new $\Phi$-entropy inequalities for diffusion semigroups, with Poincar\'e or logarithmic Sobolev inequalities as particular cases. From this study we derive the asymptotic behaviour of a large class of linear Fokker-Plank type equations under simple conditions, widely extending previous results. Nonlinear diffusion equations are also studied by means of these inequalities. The $\Gamma_2$ criterion of D. Bakry and M. Emery appears as a main tool in the analysis, in local or integral forms.Comment: 31 page

Asymptotic behaviour of reversible chemical reaction-diffusion equations

We investigate the asymptotic behavior of the a large class of reversible chemical reaction-diffusion equations with the same diffusion. In particular we prove the optimal rate in two cases : when there is no diffusion and in the classical "two-by-two" case

Super-Poincar\'e and Nash-type inequalities for Subordinated Semigroups

We prove that if a super-Poincar\'e inequality is satisfied by an infinitesimal generator $-A$ of a symmetric contracting semigroup then it implies a corresponding super-Poincar\'e inequality for $-g(A)$ with any Bernstein function $g$. We also study the converse statement. We deduce similar results for the Nash-type inequality. Our results applied to fractional powers of $A$ and to $\log(I+A)$ and thus generalize some results of Biroli and Maheux, and Wang 2007. We provide several examples.Comment: submitted. 23p. no figure. Title slightly changed. Results and text improve

Modified logarithmic Sobolev inequalities and transportation inequalities

We present a class of modified logarithmic Sobolev inequality, interpolating between Poincar\'e and logarithmic Sobolev inequalities, suitable for measures of the type \exp(-|x|^\al) or more complex \exp(-|x|^\al\log^\beta(2+|x|)) (\al\in]1,2[ and \be\in\dR) which lead to new concentration inequalities. These modified inequalities share common properties with usual logarithmic Sobolev inequalities, as tensorisation or perturbation, and imply as well Poincar\'e inequality. We also study the link between these new modified logarithmic Sobolev inequalities and transportation inequalities

Dimensional contraction via Markov transportation distance

It is now well known that curvature conditions \`a la Bakry-Emery are equivalent to contraction properties of the heat semigroup with respect to the classical quadratic Wasserstein distance. However, this curvature condition may include a dimensional correction which up to now had not induced any strenghtening of this contraction. We first consider the simplest example of the Euclidean heat semigroup, and prove that indeed it is so. To consider the case of a general Markov semigroup, we introduce a new distance between probability measures, based on the semigroup, and adapted to it. We prove that this Markov transportation distance satisfies the same properties for a general Markov semigroup as the Wasserstein distance does in the specific case of the Euclidean heat semigroup, namely dimensional contraction properties and Evolutional variational inequalities