Stefano Olla (Cergy-Pontoise): Presentation

Viviane Baladi (Orsay): Finite-dimensional functional analysis applied to transfer operators for infinite dimensional maps

We describe a simple approach to perturbative analysis of Perron-Frobenius operators, and study of the Floquet spectrum of the transfer operators of weakly coupled analytic maps on an infinite lattice (Baladi-Rugh, 2001, on mp_arc): we are able to go beyond the first spectral gap and to exhibit smooth curves of eigenvalues and eigenvectors (as functions of the crystal momenta).

Elise Janvresse (Rouen): Spectral gap for Kac's model of Boltzmann's equation

We consider the random walk on S^n-1(1), the (n-1)-dimensional sphere of radius 1, generated by random rotations on randomly selected coordinate planes i, j with 1<= i < j <= n. This dynamics was used by M. Kac as a model for the spatially homogeneous Boltzmann equation.
If we assume that the initial distribution is of product form, Kac proved that this property remains valid for all time in the limit n going to infinity. In modern terminology, Kac proved the ``propagation of chaos''. Once propagation of chaos is proved, it is straightforward to show that the marginal density of a particle satisfies an analog of a Boltzmann equation.
The spectral properties of the collision operator of the Boltzmann equation is of critical importance to understand it. Since this collision operator is generated by the process described above, a very basic property is the size of the spectral gap, which Kac conjectured to be of order 1 / n.
We prove Kac's conjecture by supplying a lower bound of the form c / n to the spectral gap. This is achieved by using the martingale method developped by Lu and Yau.

Werner Krauth (Paris): Absence of Thermodynamic Phase Transition in a Model Glass Former

The glass transition can simply be defined as the point at which the viscosity of a structurally disordered liquid reaches 10¹³ Poise. This operational definition sidesteps the principal controversy about glasses: Is the transition a purely dynamical phenomenon or is an underlying thermodynamic phase transition responsible for the dramatic slowdown at the liquid-glass boundary?
Modern Monte Carlo algorithms allow to study hard disk systems far within their glassy phase, where usual methods suffer from ergodicity breaking. For a model system, we show that indications of a thermodynamic transition are lacking up to very high densities, and that the glass is indistinguishable from the liquid on purely thermodynamic grounds. The issue of convergence time/ergodicity breaking in Monte Carlo algorithm thus mirrors one of the important enigmas in statistical physics: the nature of the glassy phase.

Michel Ledoux (Toulouse): Hypercontractivity of Hamilton-Jacobi equations

Following the equivalence between logarithmic Sobolev inequalities and hypercontractivity showed by L. Gross, we prove that logarithmic Sobolev inequalities are related similarly to hypercontractivity of solutions of Hamilton-Jacobi equations. By the infimum-convolution description of the Hamilton-Jacobi solutions, this approach provides a clear view of the connection between logarithmic Sobolev inequalities and transportation cost inequalities investigated recently by F. Otto and C. Villani. In particular, we recover in this way transportation from Brunn-Minkowski inequalities and for the exponential measure. This is joint work with S. Bobkov (University of Minnesota) and I. Gentil (University of Toulouse).

Carlangelo Liverani (Rome): Computing the rate of decay of correlations in expanding and hyperbolic systems

I will present some results in the theory of dynamical systems that allow, in certain cases, an accurate computation of the rate of convergence of equilibrium for all times.

Fabio Martinelli (Rome): Relaxation to equilibrium of Kawasaki dynamics for lattice gases

We consider a conservative stochastic spin exchange dynamics reversible with respect to the canonical Gibbs measure of a lattice gas model. We show how the relaxation time to equilibrium of a box of side L grows with L both when the corresponding grand canonical measure satisfies a suitable strong mixing condition and at low temperature.

Andrew Stuart (Warwick): Geometric ergodicity for degenerate stochastic differential equations and approximations

The ergodicity of SDEs is studied by use of techniques from the theory of Markov chains on general state spaces. Careful application of these Markov chain results leads to straightforward proofs of ergodicity for a variety of SDEs, in particular for problems with degenerate noise and for problems with locally Lipschitz vector fields. The key points which need to be verified are the existence of a Lyapunov function inducing returns to a compact set, a uniformly reachable point from within that set and some smoothness of the probability densities. Applications include the Langevin equation, the Lorenz equation with degenerate noise and gradient systems. The ergodic theorems proved are quite strong, yielding exponential convergence of expectations for classes of measurable functions restricted only by the condition that they grow no faster than the Lyapunov function. Very similar techniques allow a variety of time-discrete approximations to be studied and geometric ergodicity proved.

Denis Talay (Nice): Convergence exponentielle de moments de solutions ergodiques d'equations differentielles stochastiques, et applications numeriques

On s'interesse au comportement en temps long de certains processus de diffusion ergodiques, en particulier des solutions de systemes hamiltoniens stochastiques dissipatifs a coefficients non globalement lipschitziens. On montre que les moments de telles solutions convergent exponentiellement vite vers les moments de la probabilite invariante. Ces resultats permettent de decrire finement les erreurs d'approximation numerique de ces moments : erreur due a la discretisation du systeme differentiel stochastique, et erreur statistique due a la mise en oeuvre numerique du theoreme ergodique. Quelques exemples d'application et une extension en cours seront decrits.

Cedric Villani (Lyon): Quantitative versions of Boltzmann's H theorem

Boltzmann's H theorem states that the H-functional, or (neg)entropy, is nonincreasing for systems of particles evolving according to Boltzmann's equation. This is usually advocated as the most convincing argument for the solution to converge towards a Maxwellian distribution. Proving this convergence in full generality is still too difficult a problem nowadays, but partial results have been obtained thanks to (1) a better understanding of the mathematical structure of the entropy dissipation functional; (2) a new analysis of the degeneracy linked to the non-dissipativity of the transport term. In joint works with Toscani and with Desvillettes, I have shown that the problem of trend to equilibrium can be all reduced to a problem of uniform a priori bounds; I shall describe the corresponding tools and results.