ANR "Inegalite fonctionelles : probabilites et equations aux derivees partielles

The project will be sponsored by the ANR project and the "Pierre de Fermat" grant attributed to Boguslaw Zegarlinski for the year 2006 by the "Region Midi-Pyrénées".

Participants: D. Bakry (Toulouse), F. Barthe (Toulouse), J.P. Bartier (Paris), F. Baudoin (Toulouse), G. Blower (Lancaster), S. Bobkov (Minnesota), F. Bolley (Toulouse), P. Cattiaux (Paris), D. Chafai (Toulouse), T. Delmotte (Toulouse), J. Demange (Toulouse), D. Dizdar (Bonn), H. Djellout (Clermont-Ferrand), J. Dolbeault (Paris), P. Fougères (Paris), I. Gentil (Paris), M. Gourcy (Clermont-Ferrand), N. Gozlan (Paris), A. Guillin (Paris), C. Jimenez (Paris), A. Joulin (La Rochelle), M. Ledoux (Toulouse), C. Léonard (Paris), P. Laurencot (Toulouse), P. Lugiewicz (London), F. Malrieu (Rennes), B. Nazaret (Paris), Z. Qian (Oxford), E. Rio (Paris), J.M. Schlenker (Toulouse), K.T. Sturm (Bonn), C. Villani (Lyon), D. Wrzosek (Varsovie), L. Wu (Clermont-Ferrand), L. Xu (London), B. Zegarlinski (London), P.A. Zitt (Paris).

	Monday 3d	Tuesday 4th	Wenesday 5th
9h-10h	ZEGARLINSKI	STURM	BOBKOV
10h-10h30	Coffee	Coffee	Coffee
10h30-11h	Lugiewicz	Rio
11h10-11h40	Fougères	Gourcy	Qian
11h50-12h20	Blower	Nazaret	Wu

14h-15h	DOLBEAULT	VILLANI	CATTIAUX
15h-15h30	Coffee	Coffee	Coffee
15h30-16h	Guillin	Gentil	Leonard
16h10-16h40	Gozlan	Chafai	Bolley
16h50-17h20	Bartier	Malrieu	Joulin


19h		Buffets//Posters

Slides: Bartier, Blower, Bolley, Dolbeault, Fougères, Gentil, Gourcy, Joulin, Leonard, Nazaret

Talk titles and abstracts:

G. Blower: "Transportation Inequalities with Applications to Random matrices"
                Abstract: The talk begins with a discussion of the classical Monge transportation problem, and how it is solved through monotonic inducing maps. Talagrand's transportation inequalities bound the transportation cost for suitable cost functions by relative entropy. Some transportation inequalities apply to measures $e^{-V(x)}\,dx$ where $V:{\Bbb R}^n\rightarrow {\Bbb R}$ is a uniformly convex function. With a view to applications in random matrix theory, this talk presents the notion of displacement convexity which deals with potentials with logarithmic interaction terms. All of the results are concerned with the concentration of measure effect for measures on phase spaces of high dimension.
S. Bobkov: "Isoperimetric and Sobolev type inequalities for convex probability measures with heavy tails"
                Abstract: We are discussing the family of convex probability measures beyond the classical case of log-concave distributions. In particular, we consider the isoperimetric problem and related analytic inequalities.
F. Bolley: "Particle approximation of a mean field equation"
                Abstract: We consider the approximation of the solution to a mean field equation by a system of interacting particles. We derive non asymptotic deviation bounds of the empirical measure of these particles around the law to be approached.
D. Chafaï: "Remarks on continuous and discrete space inequalities"
               Abstract: The lack of chain rule for discrete space Markov processes blocks the use of well known diffusion tools for the derivation of functional inequalities. The goal of this short talk is to explain how convexity arguments may serve as an alternative in certain situations.
J. Dolbeault: "Free energy estimates for the two-dimensional Keller-Segel model"
                Abstract : In the two-dimensional euclidean space, free energy estimates based on the logarithmic Hardy-Littlewood-Sobolev inequality are sharp and can be used to distinguish two regimes corresponding to blow-up in finite time or global existence. In the second case, non trivial intermediate asymptotics are found using entropy type estimates.
P. Fougères: "emilinear problems, smoothing properties and functional inequalities"
                Abstract : We deal with different kinds of semilinear problems where functional inequalities ensure nice properties of the solution (and also existence). First, Sobolev type inequalities give smoothing properties of the linear part of the equation from which general conditions to avoid blow up may follow. Secondly, these inequalities "carry" related nonlinear Cauchy problems whose solutions define nonlinear Markovian type semigroups that behave surprisingly close to the linear ones.
I. Gentil: "Modified logarithmic Sobolev inequalities for log-concave measures"
                Abstract: We prove modified logarithmic Sobolev inequalities for log-concave measures in two cases. The first one is between Poincaré and logarithmic Sobolev inequalities and the second one is under the gamma_2 criterion (after logarithmic Sobolev inequality).
M. Gourcy: "Logarithmic Sobolev Inequalities of Diffusions for the L2 metric"
                Abstract : Under the Bakry-Emery's Gamma2-minoration condition, we establish the logarithmic Sobolev inequality for the Brownian motion with drift in the metric L2 instead of the usual Cameron-Martin metric. This inequality provides us the gaussian concentration inequalities for the large time behavior of the diffusion.
N. Gozlan: "Characterization of Talagrand's inequality on the real line"
                Abstract: Using a perturbation method, we prove a necessary and sufficient condition for Talagrand's transportation-cost inequality on the real line.
A. Joulin: "Poisson-type deviation bounds for curved birth-death processes"
                Abstract: In this talk, we present Poisson-type deviation bounds for curved birth-death processes, that extend the results in null curvature of C. Ané and M. Ledoux. The key point is to provide some conditions on the rates of the associated generator under which the curvatures of the semigroup are bounded below. The examples of the Ehrenfest chain and the M/M/1 queue are investigated in detail."
C. Léonard: "A large deviation approach to some transport inequalities"
P. Lugiewicz: "Ergodic property of Hörmander diffusions in infinite dimension"
                Abstract: We analyse the lattice system with the (compact) spin space being a smooth manifold equipped with a H\"ormander system of fields. Assuming that the (finite range) interaction between spins satisfies an appropriate strong mixing condition the corresponding Gibbs measure fulfils the Log-Sobolev inequality.
F. Malrieu: "Logarithmic Sobolev Inequalities for Inhomogeneous Semigroups"
B. Nazaret: "Optimal Sobolev trace inequalities on the half space"
               Abstract: Using a mass transportation method, we study optimal Sobolev trace inequalities on the half space and prove a conjecture made by Escobar in 1988 about the minimizers.
E. Rio: "Sur la vitesse de convergence dans le TLC pour les distances de Wasserstein"
               Abstract: Soit $X_1, X_2, \dots$ une suite de variables al\'eatoires r\'eelles ind\'ependantes et \'equi\-distribu\'ees de moyenne nulle et de variance 1. Soit $S_n = X_1 + X_2 + \dots + X_n$, $\mu_n$ la loi de $n^{-1/2} \, S_n$ et $\gamma$ la loi normale $N(0,1)$. Le TLC classique ainsi que l'uniforme int\'egrabilit\'e de la suite $(S_n^2 / n)_n$ assurent que la loi $\mu_n$ converge vers $\gamma$ pour les m\'etriques de Wasserstein $W_r$ d'ordre $r$ pour tout $r$ dans $[1,2]$. Esseen (1958) a montr\'e que $W_1(\mu_n, \gamma) = O(n^{-1/2})$ d\`es que ${\mathbb E}(\vert X_1 \vert^3) < \infty$. Nous \'etendons le r\'esultat d'Esseen aux r\'eels $r$ de $]1,2]$, en montrant que $W_r(\mu_n, \gamma) = O(n^{-1/2})$ d\`es que ${\mathbb E}(\vert X_1 \vert^{r+2}) < \infty$. La d\'emonstration s'appuie sur une comparaison peu connue entres les distances de Wasserstein et distances id\'eales de Zolotarev.
C. Villani: "Convergence to equilibrium for hypocoercive linear diffusions: Hormander meets log Sobolev"
L. Wu: "A $\Phi$-entropy contraction inequality for Gaussian vectors"
B. Zegarlinski: "Linear and nonlinear phenomena in large interacting systems"

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