Chemotaxis is a phenomenon by which a population of cells moves under the stimulation of a chemical signal present in its environment. At the macroscopic level, this phenomenon is modeled by the Keller-Segel system, the particularity of which is the fact that its solutions may blow-up in finite time. Motivated by the study of the fully parabolic version of the Keller-Segel model using probabilistic methods, we propose a stochastic particle system with an unusual interaction: each particle interacts with the past of all the others through a highly singular space-time kernel. We will show the existence and propagation of chaos for this system in the one-dimensional case. We will discuss the numerical results in the two-dimensional case and state an existence and uniqueness result for the non-linear SDE in the McKean Vlasov sense under explicit conditions on the model parameters in the multi-dimensional setting.
The talk will be based on several works among which one in collaboration with J-F. Jabir (HSE, Russia) and D. Talay (Inria, France).