Sur un ensemble fini, on s'intéresse aux minorations linéaires du trou spectral d'un noyau markovien réversible, en terme de la constante isopérimétrique associée. On montre que la constanteoptimale est l'inverse du cardinal de l'ensemble moins un, mais on verra aussi comment il est possible de l'améliorer dans certaines situations particulières (arbres pointés radiaux à nombre fini de générations). Une application des inégalités précédentes est de retrouver immédiatement le comportement à basse température des trous spectraux associés aux algorithmes de recuit simulé généralisés finis. Une approche similaire permet également d'obtenir très simplement une caractérisation de l'énergie virtuelle due à Trouvé.

This paper gives an approximation of the solution of the Boltzmann equation by stochastic interacting particle systems in a case of cut-off collision operator and small initial data. In this case,following the ideas of Mischler and Perthame, we prove the existence and uniqueness of the solution of this equation and also the existence and uniqueness of the solution of the associated nonlinear martingale problem. 
Then, we first delocalize the interaction by considering a mollified Boltzmann equation in which the interaction is averaged on cells of fixed size which cover the space. In this situation, Grahamand Méléard have obtained an approximation of the mollified solution by some stochastic interacting particle systems. Then we consider systems in which the size of the cells depends on the size of the system. We show that the associated empirical measures converge in law to a deterministic probability measure whose density flow is the solution of the full Boltzmann equation. That suggests an algorithm based on the Poisson interpretation of the integral term for the simulation of this solution.

A multidimensional version of the results of Komlos, Major and Tusnady for the Gaussian approximation of the sequence of successive sums of independent random vectors with finiteexponential moments is obtained.

The problem of predicting integrals of stochastic processes is considered. Linear estimators have been constructed by means of samples at N discrete times for processes having a fixedHölderian regularity s > 0 in quadratic mean. It is known that the rate of convergence of the mean squared error is of order N-(2s+1). In the class of analytic processes H p , p ≥ 1, we show that among all estimators, the linear ones are optimal. Moreover, using optimal coefficient estimators derived through the inversion of the covariance matrix, the corresponding maximal error has lower and upper bounds with exponential rates. Optimal simple nonparametric estimators with optimal sampling designs are constructed in H² and H ∞ and have also bounds with exponential rates.

We prove a full large deviations principle, in the scale N2, for the empirical measure of the eigenvalues of an N x N (non self-adjoint) matrix composed of i.i.d. zero mean random variables with variance N-1. The (good) rate function which governs this rate function possesses as unique minimizer the circular law, providing an alternative proof of convergence to the latter. The techniques are related to recent work by Ben Arous and Guionnet, who treat the self-adjoint case. A crucial role is played by precise determinant computations due to Edelman and to Lehmann and Sommers.

We study the stochastic homogenization processes considered by Baldi (1988) and by Facchinetti and Russo (1983). We precise the speed of convergence towards the homogenized state by proving the following results: (i) a large deviations principle holds for the Young measures; if the Young measures are evaluated on a given function, then (ii) the speed of convergence is bounded in every Lp norm by an explicit rate and (iii) central limit theorems hold. In dimension 1, we apply these results to the stochastic homogenization of random p-Laplacian operators for any p > 1.

We prove a strong law of large numbers for moving averages of independent, identically distributed random variables with certain subexponential distributions. These random variables show a behavior that can be considered intermediate between the classical strong law and the Erdös-Rényi law. We further show that the difference from the classical behavior is due to the influence of extreme terms.