Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - Université de Rouen Normandie

Nous étudions des milieux composites 2D formés d’inclusions régulières
plongées dans une phase matrice. Nous analysons comment le gradient de la
solution d’une EDP elliptique scalaire explose lorsque des inclusions sont presque
en contact, en fonction de la distance inter-inclusion et du contraste des conductivités.

Le temps de mélange d’une marche aléatoire sur un graphe connexe fini est intimement lié à la présence de goulots d’étranglement (« bottlenecks ») dans le graphe: intuitivement, plus il est difficile pour la marche de s’échapper de certaines régions du graphe, plus le mélange est lent. De plus, la présence de goulots d’étranglement étroits empêche souvent le cutoff, qui décrit une convergence abrupte à l’équilibre.

On fixe la forme d'un diagramme de Young et on tire au hasard un remplissage de ce diagramme : on numérote les cases de manière à être croissant suivant les lignes et les colonnes. On obtient un objet qu'on peut voir comme une surface aléatoire. On étudie certaines propriétés de ces surfaces dans le cas d'un diagramme rectangulaire et triangulaire. On montre en particulier qu'on peut obtenir comme loi limite, en divers endroits du diagramme, la loi gaussienne, Tracy Widom ou des lois apparentées aux lois de Mittag-Leffler.

The objective of segmentation methods is to detect abrupt changes, called breakpoints, in the distribution of a signal. Such segmentation problems arise in many areas, as in biology, in climatology, in geodesy, .... The inference of segmentation models requires to search over the space of all possible segmentations, which is prohibitive in terms of computational time, when performed in a naive way.

In multivariate kernel density estimation, the bandwidth selection remains a challenge in terms of algorithmic performance and quality of the resulting estimation. A recently developped method, the Penalized Comparison to Overfitting (PCO), is compared to other usual bandwidth selection methods for multivariate and univariate kernel density estimation. In particular, the cross-validation and plug-in estimators are numerically investigated and compared to PCO.

We consider two models of Markov processes observed (continuous time) in white Gaussian noise. One is telegraph signal with two states (ergodic case) and the second is partially observed linear system with small noise in observation equation. For both models we construct the MLE and One-step MLEs of the unknown parameters and show the consistency, asymptotic normality, convergence of moments and the asymptotic minimax efficiency of these estimators.

This work deals with multi state systems that we model by means of semi-Markov processes. The sojourn times are seen to be independent not identically distributed random variables and assumed to belong to a general class of distributions that is closed under maxima. We obtain maximum likelihood estimators of the parameters of interest and investigate their asymptotic properties. Plug-in type estimators are furnished for various reliability quantities related to the system under study.

I will talk about recent progress related to nonsingular Bernouli shifts. Properties of conservativeness, ergodicity, Krieger's type, K-property will be under discussion. "General shifts", i.e. those which are not the natural extensions of nonsingular endomorphisms, will be also considered

We consider the problem of estimating function in a periodic regression in continuous time with the Levy noise by discrete time observations. We use the model selection approach and develop a new adaptive procedure, which involves special modifications of the well-known James-Stein estimates, to improve the accuracy of the basic weighted least square estimates.