Workshop
"FUNCTIONAL INEQUALITIES AND DISCRETE SPACES"


January 1114, 2011


to be held in MarnelaVallée


The registration is free but mandatory (the lunch will be offered if you register).





Shortcourses (2 hours)  Lectures (25 minutes)  
Nathaël Gozlan (ParisEst Marne la Vallée)  Sergey Bobkov (Minnesota)  
Aldéric Joulin (Toulouse)  Anca Bonciocat (IMAR, Bucharest)  
Yao Li and HaoMin Zhou (Georgia Tech.)  Pietro Caputo (Roma 3)  
Emanuel Milman (Technion, Haifa)  Patrick Cattiaux (Toulouse)  
Yann Ollivier (Paris Sud)  Dario CorderoErausquin (Paris 6)  
Prasad Tetali (Georgia Tech.)  Ivan Gentil (Lyon 1)  
Arnaud Guillin (Clermont)  
Vasileios Kontis (Imperial College of London)  
Christian Léonard (parisOuest)  
Joaquin Martin (Universitat Autonoma de Barcelona) 
Tuesday 11 
Wednesday 12 
Thursday 13 
Friday 14 
10h0010h30. AccueilCoffee 
9h4510h45.
P. Tetali1:
Isoperimetric and Functional inequalities in
discrete spaces 
9h4510h45.
N. Gozlan1 :
Concentration of measure and
optimal transport 
9h4510h45.
Y. Li/H. Zhou1 :
FokkerPlanck Equations for a Free Energy Functional
or Markov Process on a graph 
10h3011h30.
Y. Ollivier1 :
Discrete and continuous concentration:
A unified approach 
10h4511h15. Coffee 
10h4511h15. Coffee 
10h4511h15. Coffee 
11h3011h45. Break 
11h1512h15.
P. Tetali2 :
Isoperimetric and Functional inequalities
in discrete spaces 
11h1512h15.
N. Gozlan2 :
Concentration of measure and
optimal transport 
11h1512h15.
Y. Li/H. Zhou2 :
FokkerPlanck Equations for a Free Energy Functional
or Markov Process on a graph 
11h4512h45.
Y. Ollivier2:
Discrete and continuous concentration: A unified approach 
12h1512h30. Break 
12h1512h30. Break 
12h1512h30. Break 

12h3012h55.
A. Bonciocat :
Curvature bounds: discrete versus continuous spaces

12h3012h55.
P. Caputo :
Zero temperature 3D stochastic Ising model:
a first step towards motion by mean curvature 
12h3012h55.
J. Martin :
Pointwise symmetrization inequalities and
Sobolev inequalities on Probability Metric Spaces 
12h4514h30. Lunch 
12h5514h30. Lunch 
12h5514h30. Lunch 
12h5514h30. Lunch 
14h3015h30.
A. Joulin1 :
Curvatures and intertwinings of
birthdeath processes 
14h3014h55.
P. Cattiaux :
Hitting times and
Poincaré Inequalities 
14h3015h30.
E. Milman1 :
Isoperimetric Inequalities:
Methods and Applications 
14h3014h55.
D. Cordero :
Symmetries in variance estimates

15h3015h45. Break 
15h0015h25.
A. Guillin:
Weighted logarithmic Sobolev inequalities 
15h3015h45. Break 
15h0015h25.
V. Kontis:
An inequality of Ledoux for log concave probability measures 
15h4516h45.
A. Joulin2 :
Curvatures and intertwinings of
birthdeath processes 
15h3015h55.
I. Gentil :
Dimension dependent hypercontractivity
for Gaussian kernels 
15h4516h45.
E. Milman2 :
Isoperimetric Inequalities:
Methods and Applications 
15h3015h55.
S. Bobkov :
Concentration of information in data generated
by logconcave probability distributions 

16h0016h25.
C. Léonard :
Random displacement interpolations 


The aim of this course is to present recent results on transport inequalities obtained in collaboration with C. Léonard, C. Roberto and PM Samson. In the first part, I will give a necessary and sufficient condition for Talagrand's inequality on the real line. In the second part, I will explain the links between Talagrand's inequality and the dimension free Gaussian concentration phenomenon. This will lead us to a new proof of OttoVillani Theorem on an arbitrary polish metric space. Finally, in the third part we will show the equivalence, in a general metric space framework, between Talagrand's inequality and a variant of the LogSobolev inequality. This theorem will enable us to prove a general perturbation result for Talagrand's inequality.
In these lectures we focus on integervalued Markov processes (birthdeath processes). The main purpose of this talk is to show how the notion of discrete curvature recently developed by Y. Ollivier is related, in our birthdeath framework, to intertwining relations between gradients and convenient semigroups.
After a brief review of the curvature and its standard properties, we recall on the one hand the coupling method used by M.F. Chen to calculate the curvature of a given birthdeath process (with respect to a large class of distances). On the other hand, we show how to recover this result by providing a simple intertwining relation betwen a given discrete gradient and two different semigroups: the first one is that of the original birthdeath process whereas the second one is a FeynmanKac (FK) semigroup, whose potential is nothing but the curvature of interest. The method emphasized in the proof is remarkably simple and rely mainly on interpolation and convexity. Various applications of the intertwinings are provided in terms of (optimal) functional inequalities : Poincare, modified logSobolev and transportationinformation inequalities, weighted isoperimetry, etc... This talk is based on the works of Y. Ollivier (curvature), M.F. Chen (coupling) and a joint work with D. Chafai (intertwinings).
FokkerPlanck equation is a linear parabolic equation which describes the time evolution of probability distribution of a stochastic process defined on a Euclidean space. Corresponding to the stochastic process, there often is a free energy functional which is defined on the space of probability distributions and is a linear combination of a potential and an entropy. In recent years, it has been shown that FokkerPlanck equation is the gradient flow of the free energy functional defined on the Riemannian manifold of probability distributions whose inner product is generated by a Wasserstein distance. In this talk, we will consider similar matters for a free energy functional or Markov process defined on a graph with a finite number of vertices and edges. If N is the number of vertices of the graph, then we will show that the corresponding FokkerPlanck equation is a system of N nonlinear ordinary differential equations defined on a Riemannian manifold of probability distributions. However, in contrast to the case of stochastic processes defined on Euclidean spaces, we have different choices for inner products for the set of probability distributions resulting in different FokkerPlanck equations for the same process with different random perturbations. It is shown that there is a strong connection but also substantial differences between the ordinary differential equations and the usual FokkerPlanck equation on Euclidean spaces. Furthermore, the Riemannian manifold in our definition has many similar properties with the 2Wasserstein space, which are related to the gradient flow, the geodesic and the optimal transport. Some examples will also be discussed.
In the first part, we survey some old and recent methods for obtaining dimensionindependent isoperimetric inequalities on Riemannian manifolds equipped with a density, under a suitable lower bound on the (generalized) Ricci curvature of the space. We will focus on the BakryÉmeryLedoux semigroup method, Gromov's geometric method, isoperimetric profile method, and time permitting, contraction methods. In particular, the connection to concentration inequalities, Sobolevtype inequalities and possibly TransportEntropy inequalities will be emphasized.
In the second part, we describe various applications of these dimensionindependent connections: The first pertains to the stability of isoperimetric, logSobolev, Poincaré and TransportEntropy inequalities under perturbation of the underlying measure in the presence of a curvature lowerbound. The extent of the perturbation is measured using several different distances between perturbed and original measure, such as a onesided L_\infty bound on the ratio between their densities, TotalVariation, Wasserstein distances, and relative entropy (KullbackLeibler divergence). In particular, we extend the classical HolleyStroock perturbation lemma to this setting, improving the dependence on the perturbation parameter from linear to logarithmic.
Next, we show the equivalence of TransportEntropy inequalities with different costfunctions in the presence of a curvature lower bound. Time permitting, we will demonstrate how to tighten weak TransportEntropy inequalities in this setting.
Lastly, in the compact setting, we obtain an optimal (up to numeric constants) isoperimetric inequality as a function of the curvature lower bound and diameter upper bound. In particular, the best known Cheeger and logSobolev inequalities are obtained in this setting.
We will present a criterion inspired by positive curvature in differential geometry, which allows for a unified treatment of concentration of measure on both continuous and discrete spaces. This criterion, first introduced by Dobrushin in 1970 in a very different context, can actually be interpreted as a curvature in discrete spaces.
Applications range from new and often nearoptimal concentration inequalities for Monte Carlo Markov chain simulation techniques (work with Alderic Joulin), such as waiting queues, to improvement in classical estimates for differential geometry, such as the spectral gap the LaplaceBeltrami operator (work of Laurent Veysseire).
In the twopart lecture, I will attempt to describe connections between various functional inequalities in a discrete framework. The inequalities include logarithmic and modified logarithmic Sobolev inequalities, Subgaussian and Transportation inequalities.
Refinements such as Isoperimetric and Spectral profile as well as recent generalizations of the classical CheegerMazya type isoperimetric inequalities will be described in the second lecture. Time permitting, I will touch upon smallset expansion, semidefinite relaxation of spectral profile and other algorithmic aspects of the profile. Some examples and open questions will also be mentioned.
Let $X$ be random vector with logconcave density $f(x)$ on $R^n$. The information content is defined as the random variable $log f(X)$. We will be discussing concentration properties of the distributions of such random variables uniformly over the class of all logconcave measures on the Euclidean spaces of high dimensions. (Joint work with M. Madiman)
We present rough (approximate) lower curvature bounds and rough curvaturedimension conditions for discrete spaces and for graphs. We derive perturbed transportation cost inequalities, that imply mass concentration and exponential integrability of Lipschitz maps. For spaces that satisfy a rough curvaturedimension condition we obtain a generalized BrunnMinkowski inequality and a BonnetMyers type theorem. (Based on a joint work with K.T. Sturm)
Recent developments on the statistics of monotone surfaces and dimer coverings of the honeycomb lattice are used to analyze convergence to equilibrium for the 3D zero temperature stochastic Ising model with "+" boundary conditions started from an all "" initial configuration. In particular, the time needed for an initial droplet of linear size L to disappear is shown to satisfy upper and lower bounds of order L^2 up to logarithmic corrections. This is joint work with Fabio Martinelli, Fabio Lucio Toninelli, and Francois Simenhaus.
Joint work with Guillin, Zitt.
We investigate how to use, in the usual L2 approach, the symmetries (by isometries) of a distribution in order to get bounds for the variance or for the spectral gap. Applications to logconcave distributions (and convex bodies) and to the spectral gap of conditioned spin system will be given.
We derive sharp, local and dimension dependent hypercontractive bounds on the Markov kernel of a large class of diffusion semigroups. Unlike the dimension free ones, they capture refined properties of Markov kernels, such as trace estimates.
we propose here, via Lyapunov conditions, an easy way to derive weighted logarithmic Sobolev inequalities, allowing to find the optimal (in the one dimensional case) weight. Some applications are discussed.
We prove a functional inequality introduced by M. Ledoux in 1988 for a class of probability measures whose density satisfies a certain convexity condition. Part of the motivation comes from the isoperimetric content of the inequality as well as the fact that it implies a logarithmic Sobolev inequality. The method applies to a number of cases including the one where the underlying space is a Lie group of Heisenberg type. This is joint work with J. Inglis and B. Zegarlinski.
It is known that displacement interpolation based on quadratic optimal transport is a useful tool to derive some functional inequalities, e.g. logarithmic Sobolev inequalities or Talagrand transport inequalities. We describe a similar object which is defined in terms of stochastic processes, where the Markov dynamics plays the role of a transport cost function. This approach seems to be flexible enough to be applied to discrete spaces. This is a work in progress.
To formulate Sobolev inequalities one needs to answer questions like: what is the role of dimension? What norms are appropriate to measure the integrability gains? etc. For example, in contrast to the Euclidean case, the integrability gains in Gaussian measure are logarithmic but dimension free (log Sobolev inequalities). I will discuss some methods to prove general Sobolev inequalities that unify the Euclidean and the Gaussian cases, as well as several important model manifolds. This is joint work with Mario Milman.