Inserted: 31 jul 2017
Last Updated: 18 jun 2018
Journal: ESAIM Control Optim. Calc. Var.
This paper presents a class of evolutive Mean Field Games with multiple solutions for all time horizons $T$ and convex but non-smooth Hamiltonian $H$, as well as for smooth $H$ and $T$ large enough. The phenomenon is analysed in both the PDE and the probabilistic setting. The examples are compared with the current theory about uniqueness of solutions. In particular, a new result on uniqueness for the MFG PDEs with small data, e.g., small $T$, is proved. Some results are also extended to MFGs with two populations.
Keywords: uniqueness of solutions, Mean Field Games, finite horizon, non-uniqueness of solutions, multipopulation MFG