On a probabilistic interpretation of the parabolic-parabolic Keller Segel equations
par
Milica Tomasevic(Ecole Polytechnique)
→
Europe/Paris
Salle de Séminaires (Orléans)
Salle de Séminaires
Orléans
Description
The Keller Segel model for chemotaxis is a two-dimensional system of parabolic or elliptic PDEs. Its particularity is that the solutions may blow-up in finite time. Motivated by the study of the fully parabolic model using probabilistic methods, we give rise to a non linear stochastic differential equation of McKean-Vlasov type with a highly non standard and singular interaction. Indeed, the drift of the equation involves all the past of one dimensional time marginal distributions of the process in a singular way. In terms of approximations by particle systems, an interesting and challenging difficulty arises: at each time each particle interacts with all the past of the other ones by means of a highly singular space-time kernel.
In this talk, after reviewing the literature about the Keller-Segel model, we will derive the above probabilistic interpretation and do an overview of results obtained. Some numerical insights will also be presented.
