Résumé : For some differential equations the addition of a carefully chosen, random noise term can produce a regularising effect (e.g. solutions are more regular, or restored uniqueness).
I will first mention a few easy examples (ODEs) to introduce some of these regularising mechanisms, then detail two cases where we have regularisation for a PDE : the linear transport equation and a kinetic equation with force term. I will present some classical results for these two equations, related to well-posedness and regularity of solutions, that in the stochastic setting can be obtained under weaker hypothesis. These results are based on a careful analysis of the stochastic characteristics and the regularising properties of some associated parabolic/elliptic PDE.
If time allows, I will conclude by introducing a new strategy of proof based on stochastic exponentials and an associated parabolic PDE, which allows to obtain, under even weaker hypothesis, wellposedness for stochastic PDEs in a class of solutions which are only regular in mean. This will be illustrated by an application to the transport equation.
Lieu : Amphithéâtre Schwartz
Résumé : One of the most important cellular behaviors is cell crawling migration. It is observed
in many cellular systems both in culture and in vivo, and involved in many
essential physiological or pathological processes (wound healing, embryonic development, cancer metastasis etc). The most common mode of cell migration is lamellipodium-based migration, where speci-c adhesion points transmit intracellular pulling forces from the cytoskeleton to the substrate. In this framework, we will first present the Filament-Based Lamellipodium Model, a continuous two-dimensional, two-phase model which models the lamelipodium as a set of interconnected actin filaments interacting with each other and with the substrate through adhesion. The numerical simulations can reproduce stationary and moving steady states qualitatively similar to what is observed in real tissues for adhesion-based cell migration. As in the last decade adhesion-independent migration has been observed in confining environement and has emerged as a possibly common migration mode, the second part of the talk will be dedicated to a simpli-fied 2D model recently developed for focal adhesion-free cell migration : A cell is modeled through its membrane represented as a set of connected springs which undergo internal pressure forces. The renewal of the actin network is modelled by creation/suppression of springs in the membrane, and we suppose that the cell generates internal counter-forces compensating mass displacement due to membrane renewal. With this very simple model, we are able to trigger cell migration, with speed depending on the geometrical characteristics of the confining environement. Our results seem to be in qualitative agreement with the biological observations.
Lieu : Salle MIP
Résumé : Le lemme de Morse est un résultat classique en courbure négative, il entraine que dans un espace CAT(-1) un rayon quasi-géodésique est à distance d’Hausdorff bornée d’un unique rayon géodésique. Récemment, Kapovich-Leeb-Porti on montré un énoncé de ce type pour les espaces symétriques. Le but de l’exposé est d’expliquer une nouvelle preuve du théorème de K-L-P. Ceci est un travail en collaboration avec J. Bochi (Santiago de Chile) et R. Potrie (Montévideo).
Lieu : Salle Picard
Notes de dernières minutes : Contre exemple de Nagata