On a probabilistic interpretation of the Keller-Segel equations of parabolic-parabolic type.

Summary In chemotaxis, the classical parabolic-parabolic Keller-Segel model in dimension d describes the time evolution of the density of a population of cells and the concentration of a chemical attractor. This thesis deals with the study of the parabolic Keller-Segel equations by probabilistic methods. To this end, we construct a nonlinear stochastic differential equation in the McKean-Vlasov sense whose drift coefficient depends, in a singular way, on the whole past of the marginal laws in time of the process. These marginal laws coupled with a judicious transformation allow to interpret the Keller-Segel equations in a probabilistic way. As far as the particle approximation is concerned, we have to overcome an interesting and, it seems to us, original and difficult difficulty : each particle interacts with the past of all the others through a strongly singular space-time kernel. In dimension 1, whatever the values of the model parameters, we prove that the Keller-Segel equations are well posed in all space and that the same is true for the corresponding McKean-Vlasov stochastic differential equation. Then, we prove the well-posedness of the associated system of non-Markovian and singular interacting particles. We also establish the propagation of the chaos to a unique mean field limit whose marginal laws in time solve the parabolic Keller-Segel system. In dimension 2, too large model parameters can lead to a finite time explosion of the solution to the Keller-Segel equations. Indeed, we show the well-posedness of the nonlinear process in the McKean-Vlasov sense by imposing constraints on the initial parameters and data. To obtain this result, we combine techniques of partial differential equation analysis and stochastic analysis. Finally, we propose a fully probabilistic numerical method to approximate the solutions of the two-dimensional Keller-Segel system and we present the main results of our numerical experiments.