Laurent Saloff-Coste

Aspects of Sobolev-type inequalities

Preface Introduction 1. Sobolev inequalities in Rn 2. Mosers elliptic Harnack Inequality 3. Sobolev inequalities on manifolds 4. Two applications 5. Parabolic Harnack inequalities.

Isopérimétrie pour les groupes et les variétés

Dans cet article, nous proposons une approche tres directe de differents inegalites isoperimetriques.

What Do We Know about the Metropolis Algorithm

The Metropolis algorithm is a widely used procedure for sampling from a specified distribution on a large finite set. We survey what is rigorously known about running times. This includes work from statistical physics, computer science, probability, and statistics. Some new results (Propositions 6.1?6.5) are given as an illustration of the geometric theory of Markov chains.

Random Walks on Finite Groups

Markov chains on finite sets are used in a great variety of situations to approximate, understand and sample from their limit distribution. A familiar example is provided by card shuffling methods. From this viewpoint, one is interested in the “mixing time” of the chain, that is, the time at which the chain gives a good approximation of the limit distribution. A remarkable phenomenon known as the cut-off phenomenon asserts that this often happens abruptly so that it really makes sense to talk about “the mixing time”. Random walks on finite groups generalize card shuffling models by replacing the symmetric group by other finite groups. One then would like to understand how the structure of a particular class of groups relates to the mixing time of natural random walks on those groups. It turns out that this is an extremely rich problem which is very far to be understood. Techniques from a great variety of different fields — Probability, Algebra, Representation Theory, Functional Analysis, Geometry, Combinatorics — have been used to attack special instances of this problem. This article gives a general overview of this area of research.

Parabolic Harnack inequality for divergence form second order differential operators

Old and recent results concerning Harnack inequalities for divergence form operators are reviewed. In particular, the characterization of the parabolic Harnack principle by simple geometric properties -Poincare inequality and doubling property- is discussed at length. It is shown that these two properties suffice to apply Mosers iterative technique.

Gibbs Sampling, Exponential Families and Orthogonal Polynomials.

We give families of examples where sharp rates of convergence to stationarity of the widely used Gibbs sampler are available. The examples involve standard exponential families and their conjugate priors. In each case, the transition operator is explicitly diagonalizable with classical orthogonal polynomials as eigenfunctions.

Nash Inequalities for Finite Markov Chains

This paper develops bounds on the rate of decay of powers of Markov kernels on finite state spaces. These are combined with eigenvalue estimates to give good bounds on the rate of convergence to stationarity for finite Markov chains whose underlying graph has moderate volume growth. Roughly, for such chains, order (diameter) steps are necessary and suffice to reach stationarity. We consider local Poincaré inequalities and use them to prove Nash inequalities. These are bounds onl2-norms in terms of Dirichlet forms andl1-norms which yield decay rates for iterates of the kernel. This method is adapted from arguments developed by a number of authors in the context of partial differential equations and, later, in the study of random walks on infinite graphs. The main results do not require reversibility.

Analyse sur les groupes de Lie à croissance polynômiale

RésuméOn déduit des estimations gaussiennes supérieures du noyau de la chaleur des estimations du même type pour les premières dérivées spatiales. On obtient ainsi des estimations gaussiennes inférieures du noyau de la chaleur. On donne des applications de ces résultats.AbstractFrom Gaussian upper bounds on the heat kernel we deduce similar upper bounds on the first space derivatives of the heat kernel. Gaussian lower bounds on the heat kernel are deduced and some applications are given.

On the stability of the behavior of random walks on groups

We show that, for random walks on Cayley graphs, the long time behavior of the probability of return after 2n steps is invariant by quasi-isometry.

Moderate growth and random walk on finite groups

We study the rate of convergence of symmetric random walks on finite groups to the uniform distribution. A notion of moderate growth is introduced that combines with eigenvalue techniques to give sharp results. Roughly, for finite groups of moderate growth, a random walk supported on a set of generators such that the diameter of the group is γ requires order γ2 steps to get close to the uniform distribution. This result holds for nilpotent groups with constants depending only on the number of generators and the class. Using Gromovs theorem we show that groups with polynomial growth have moderate growth.