Mean Field Games with state constraints: from mild to pointwise solutions of the PDE system.

20th June 2019 - h 14:00 - Room M

Mean Field Games (MFG) with state constraints are differential games with infinitely many agents, each agent facing a constraint on his state. In this case, the existence and uniqueness of Nash equilibria cannot be deduced as for unrestricted state space because, for a large set of initial conditions, the uniqueness of solutions to the minimization problem which is solved by each agent is no longer guaranteed. Therefore, we attack the problem by interpreting equilibria as measures in a space of arcs and we introduce the definition of mild solution for MFG with state constraints. More precisely, we define a mild solution as a pair (u, m) ∈ C([0, T ] × Ω) × C([0, T ]; P(Ω)), where m is given by m(t) = e_t#η for some constrained MFG equilibrium η and u(x,t) is the value function associated to the problem.
For more details see [1]. The aim of this talk is to provide a meaning of the PDE system associated with these games, the so-called Mean Field Game system with state constraints. For this, we will analyze the regularity of mild solution and we will show that it satisfies the MFG system in suitable pointwise sense. These results have been obtained in collaboration with Piermarco Cannarsa (Rome Tor Vergata) and Pierre Cardaliaguet (Paris-Dauphine).

References
[1] Cannarsa, P., Capuani, R., Existence and uniqueness for Mean Field Games with state constraints,”PDE models for multi-agent phenomena”, P. Cardaliaguet, A.Porretta, F. Salvarani editors, Springer INdAM Series, 2017.
[2] Cannarsa, P., Capuani, R., and Cardaliaguet, P., C 1,1 –smoothness of constrained solutions in the calculus of variations with application to mean field games, Mathematics in Engineering,1(1): 174-203,doi:10.3934/Mine.2018.1.174., 2018.
[3] Cannarsa, P., Capuani, R., and Cardaliaguet, P., Mean Field Games with state constraints: from mild to pointwise solutions of the PDE system, submitted, https://arxiv.org/abs/1812.11374, 2018.