19 apr 2017 -- 17:00 [open in google calendar]
Sala Seminari (Dipartimento di Matematica di Pisa)
Mean Field Game (MFG) theory is the study of strategic decision making in a very large population of small interacting individuals. We consider models where agents prefer clustering in high-density areas. The goal is to understand the existence of smooth solutions to the corresponding MFG system of coupled ergodic Hamilton-Jacobi-Bellman and Kolmogorov equations. Depending on the growth rate at infinity of the cost function, we observe two different situations: existence of solutions in a ''subcritical'' case, and non-existence in a ''supercritical'' case. This scenario is in analogy with focusing nonlinear Schrodinger equations, where the boundary between existence and non-existence of solutions is the so-called critical Sobolev exponent. We show how blow-up techniques and Pohozaev identities apply in the MFG setting. Finally, we discuss some possible generalization.