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September  2019, 14(3): 537-566. doi: 10.3934/nhm.2019021

## A class of infinite horizon mean field games on networks

 1 Univ. Paris Diderot, Sorbonne Paris Cité, Laboratoire Jacques-Louis Lions, UMR 7598, UPMC, CNRS, F-75205 Paris, France 2 Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France 3 Univ Rennes, INSA Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France

* Corresponding author: Yves Achdou

Received  July 2018 Revised  January 2019 Published  May 2019

Fund Project: The authors were partially supported by ANR project ANR-16-CE40-0015-01. The work of O. Ley and N. Tchou was partially supported by the Centre Henri Lebesgue ANR-11-LABX-0020-01

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 $\rho$. 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.

Citation: Yves Achdou, Manh-Khang Dao, Olivier Ley, Nicoletta Tchou. A class of infinite horizon mean field games on networks. Networks & Heterogeneous Media, 2019, 14 (3) : 537-566. doi: 10.3934/nhm.2019021
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Left: the network $\Gamma$ in which the edges are oriented toward the vertex with larger index ($4$ vertices and $4$ edges). Right: a new network $\tilde \Gamma$ obtained by adding an artificial vertex ($5$ vertices and $5$ edges): the oriented edges sharing a given vertex $\nu$ either have all their starting point equal $\nu$, or have all their terminal point equal $\nu$