Seminario Dottorato 2022/23 - Chiara Bernardini (29.03.2023)

In this seminar we introduce second-order ergodic Mean-Field Games systems defined in the whole space R^n with coercive potential and aggregating nonlocal coupling, defined in terms of a Riesz interaction kernel. From a PDE viewpoint, equilibria of the differential game solve a system of PDEs where an Hamilton-Jacobi-Bellman equation is combined with a  Kolmogorov-Fokker-Planck equation for the mass distribution. Due to the interplay between the strength of the attractive term and the behavior of the diffusive part, we obtain three different regimes for existence and nonexistence of classical solutions to the MFG system. 

After briefly introducing the model, we present the main ideas underlying the proof of our results. Finally, we study the behavior of solutions in the vanishing viscosity limit, namely when the diffusion becomes negligible.