Orateur
Description
In this talk, we study the convergence of the empirical measure of moderately interacting particle systems with singular interaction kernels.
Using a semigroup approach combined with a fine analysis of the regularity of infinite-dimensional stochastic convolution integrals, we prove quantitative convergence of the time marginals of the empirical measure of particle positions towards the solution of the limiting non-linear Fokker-Planck equation. The convergence rates are polynomial and involve, on one side the regularity parameters of the interaction kernel, and on the other side, the stochastic convolution integral decay rate.
Second, we prove the well-posedness for the McKean-Vlasov SDE involving such singular kernels and the convergence of the empirical measure towards it (propagation of chaos).
These results only require very weak regularity on the interaction kernel, which permits to treat models for which the classical particle system is not known to be well-defined. For instance, this includes attractive kernels such as Riesz and Keller-Segel kernels in arbitrary dimension. In particular, this convergence still holds (locally in time) for PDEs exhibiting a blow-up in finite time.
We weill tackle also some recent advances, including Burgers equation and the case of non-conservative Keller-Segel model with logistic source (involving a branching and interacting particle system).
Based on works with A. Richard (CentraleSupelec), C. Olivera (Unicamp) and T. Cavallazzi.