A Class of Infinite Horizon Mean Field Games on Networks

Yves Achdou (1) , Manh-Khang Dao (2) , Olivier Ley (2) , Nicoletta Tchou (2)

1 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
2 IRMAR - Institut de Recherche Mathématique de Rennes

We consider stochastic mean field games for which the state space is a network. In the ergodic case, they are described by a system coupling a Hamilton-Jacobi-Bellman equation and a Fokker-Planck equation, whose unknowns are the invariant measure m, a value function u, and the ergodic constant ρ. The function u is continuous and satisfies general Kirchhoff conditions at the vertices. The invariant measure m satisfies dual transmission conditions: in particular, m is discontinuous across the vertices in general, and the values of m on each side of the vertices satisfy special compatibility conditions. Existence and uniqueness are proven, under suitable assumptions.

Domaines: Equations aux dérivées partielles [math.AP]

Yves Achdou, Manh-Khang Dao, Olivier Ley, Nicoletta Tchou. A Class of Infinite Horizon Mean Field Games on Networks. Networks and Heterogeneous Media, 2019, 14 (3), pp.537-566. ⟨10.3934/nhm.2019021⟩. ⟨hal-01802120⟩