MEAN FIELD GAMES WITH CONGESTION

Yves Achdou (1) , Alessio Porretta (2)

1 LJLL - Laboratoire Jacques-Louis Lions
2 DIPMAT - Dipartimento di Matematica [Roma II]

We consider a class of systems of time dependent partial differential equations which arise in mean field type models with congestion. The systems couple a backward viscous Hamilton-Jacobi equation and a forward Kolmogorov equation both posed in (0, T) × (R N /Z N). Because of congestion and by contrast with simpler cases, the latter system can never be seen as the optimality conditions of an optimal control problem driven by a partial differential equation. The Hamiltonian vanishes as the density tends to +∞ and may not even be defined in the regions where the density is zero. After giving a suitable definition of weak solutions, we prove the existence and uniqueness results of the latter under rather general assumptions. No restriction is made on the horizon T .

Domaines: Equations aux dérivées partielles [math.AP]

Yves Achdou, Alessio Porretta. MEAN FIELD GAMES WITH CONGESTION. Annales de l'Institut Henri Poincaré, 2018, 35 (2), pp.443--480. ⟨hal-01544509⟩