Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - Université de Rouen Normandie

Dans cet exposé, nous présenterons le théorème ergodique pour des U-statistiques d'ordre deux : des conditions suffisantes pour la
convergence presque sûre et dans $L^1$ ainsi que des exemples montrant que la convergence ne peut pas avoir lieu en général. Les résultats présentés
sont ceux de l'article https://arxiv.org/abs/2302.04539, réalisé en collaboration avec Herold Dehling et Dalibor Volny.

In this talk we analyze the homogenization process of a problem studied by Dominique Blanchard, Luciano Carbone and Antonio Gaudiello. A possible approach in the case of a less regular data will also be discussed

We consider truncated detection problems for statistical models with dependent observations given by Bayesian Markov chains for a uniform prior distribution when the number of observations is limited by some known value.

The space of measures with finite second moment, when endowed with the Wasserstein distance, is a geodesic space but not a vector space. Therefore, building an adequate differential calculus is not straightforward, and currently subject of debate in the literature. We review the main definitions in use in the Hamilton-Jacobi and Mean Fields communities, with examples illustrating the nature of the objects and comments on the range and limitations of each point of view. 

We introduce a new stochastic partial differential equation with second- order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by different Gaussian noise. Such equation could be used in mathematical modeling of diffusion phenomena in medium consisting of two kinds of materials and undergoing stochastic perturbations. We prove the existence of the solution and we present explicit expressions of its covariance and variance functions.

We consider a non-local approximation of the time-dependent Eikonal equation defined on a Riemannian manifold. We show that the local and the non-local problems are well-posed in the sense of viscosity solutions and we prove regularity properties of these solutions in time and space. If the kernel is properly scaled, we then derive error bounds between the solution of the non-local problem and the one of the local problem, both in continuous-time and Forward Euler discretization. Finally, we apply these results to a sequence of random weighted graphs with $n$ vertices.

We consider an interacting particle system which models the sterile insect technique on a finite cylinder. When in contact with slow reservoirs , the hydrodynamic limit of the latter system is a set of coupled reaction diffusion equations with mixed boundary conditions, boundary conditions which depends on the slow-down rates of the reservoirs. The goal of this talk is to prove the existence of a hydrostatic limit for a class of parameters.

Cet exposé portera sur l'analyse d'équations de réaction-diffusion en domaines non bornés, s'articulant autour des concepts de survie et d'extinction de populations. Nous commencerons par dériver l'équation de la chaleur à partir de marches aléatoires à temps discrets, une méthode classiquement employée pour relier cette équation à la dispersion des individus dans un milieu. Après avoir discuté des propriétés fondamentales de la diffusion, nous introduirons différents termes de réaction pour modéliser la croissance et/ou la mortalité d'individus au cours du temps.

Abstract : fichier pdf

If we take a uniformly random lattice polygon in the square $[0,n]^2$, Bárány, Vershik and Sinai proved that the renormalized polygon converges towards a limit shape. We study the fluctuations of a random lattice polygon around that limit shape. After establishing a central limit theorem of finite-dimensional marginals of the boundary point of a lattice zonotope in any dimension, we proved a Donsker-type theorem for the boundary fluctuations, which involves a 2-dimensional Brownian bridge and a drift term that we identify as a random cubic curve.