Évènements

The systems of forward-backward stochastic differential equations driven by Brownian motion (FBSDEs for short) help us to model diffusion processes related to phenomena that involve environment perturbations. The drift coefficients constitute the descriptive part of a non-random ambient, while the Wiener processes permit us to describe the random perturbations involved into the dynamics through the diffusion terms. The systems of FBSDEs motion are linked to the nonlinear partial differential equations (PDEs) through the Feyman-Kac formulae. Therefore, the deterministic solutions can be obtained by probabilistic representations involving the stochastic processes that solve the FBSDEs.

During this talk, we deal with the numerical simulation of systems of stochastic particles ruled by FBSDEs associated with nonlinear PDEs appearing in fluid dynamics. To make this, we discretize locally in time the stochastic equations, and then we consider integration schemes of Euler-Maruyama type, together with the optimal quantization of the involved Wiener increments as an alternative to the Monte-Carlo simulation. Then we approximate the related conditional expectations over each temporal-spatial node of a computational domain with uniform discretization steps in time and space. Numerical results are presented to the case of analytic spatially-periodic exact solutions of the incompressible Navier-Stokes equations, in particular, a two-dimensional Taylor-Green vortex and three-dimensional Beltrami flows, for example an Arnold-Beltrami-Childress flow. The simulation algorithms follow from a completely probabilistic approach.

Cylindrical distributions are joint distributions of a circular variable and a linear variable, where the circular variable affects the linear variable. In this paper, we consider a class of cylindrical distributions based on the normal distribution, which have normal and angular conditional and marginal distributions. The distribution based on the normal distribution has the advantages of easy random number generation and simple Fisher information matrix. We also consider a regression model using the cylindrical distribution. Examples of estimation will be given for real data, and a new methodology of data analysis using the cylinder model will be given. Furthermore, we also discuss some potential extensions of the cylindrical distribution.

La fonction Delta d’Erdős-Hooley mesure la concentration des diviseurs d’un entier dans un intervalle dyadique. Récemment, Ford Koukoulopoulos et Tao ont amélioré l’encadrement de l’ordre moyen de cette fonction dû à Hall et Tenenbaum. Nous expliquerons les idées nouvelles de ces auteurs et expliquerons comment dans un travail en collaboration avec Gérald Tenenbaum nous avons précisé leur encadrement.