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

The standard d-dimensional parabolic--parabolic Keller--Segel model for chemotaxis describes the time evolution of the density of a cell population and of the concentration of a chemical attractant. This thesis is devoted to the study of the parabolic--parabolic Keller-Segel equations using probabilistic methods. To this aim, we give rise to a non linear stochastic differential equation of McKean-Vlasov type whose drift involves all the past of one dimensional time marginal distributions of the process in a singular way. These marginal distributions coupled with a suitable transformation of them are our probabilistic interpretation of a solution to the Keller Segel model. In terms of approximations by particle systems, an interesting and, to the best of our knowledge, new and challenging difficulty arises: each particle interacts with all the past of the other ones by means of a highly singular space-time kernel. In the one-dimensional case, we prove that the parabolic-parabolic Keller-Segel system in the whole Euclidean space and the corresponding McKean-Vlasov stochastic differential equation are well-posed in well chosen space of solutions for any values of the parameters of the model. Then, we prove the well-posedness of the corresponding singularly interacting and non-Markovian stochastic particle system. Furthermore, we establish its propagation of chaos towards a unique mean-field limit whose time marginal distributions solve the one-dimensional parabolic-parabolic Keller-Segel model. In the two-dimensional case there exists a possibility of a blow-up in finite time for the Keller-Segel system if some parameters of the model are large. Indeed, we prove the well-posedness of the mean field limit under some constraints on the parameters and initial datum. Under these constraints, we prove the well-posedness of the Keller-Segel model in the plane. To obtain this result, we combine PDE analysis and stochastic analysis techniques. Finally, we propose a fully probabilistic numerical method for approximating the two-dimensional Keller-Segel model and survey our main numerical results.