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

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

Integral monodromy groups of Kloosterman sheaves

We show that integral monodromy groups of Kloosterman $\ell$-adic sheaves of rank $n\ge 2$ on $\mathbb{G}_m/\mathbb{F}_q$ are as large as possible when the characteristic $\ell$ is large enough, depending only on the rank. This variant of Katz's results over $\mathbb{C}$ was known by works of Gabber, Larsen, Nori and Hall under restrictions such as $\ell$ large enough depending on $\operatorname{char}(\mathbb{F}_q)$ with an ineffective constant, which is unsuitable for applications. We use the theory of finite groups of Lie type to extend Katz's ideas, in particular the classification of maximal subgroups of Aschbacher and Kleidman-Liebeck. These results will apply to study reductions of hyper-Kloosterman sums in forthcoming work.Comment: 27 pages; incorporating the referees' comments. To appear in Mathematik

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