A class of infinite horizon mean field games on networks

Yves Achdou; Manh-Khang Dao; Olivier Ley; Nicoletta Tchou; Yves Achdou; Manh-Khang Dao; Olivier Ley; Nicoletta Tchou

doi:10.3934/nhm.2019021

Networks and Heterogeneous Media

2019, Volume 14, Issue 3: 537-566. doi: 10.3934/nhm.2019021

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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

Received: 01 July 2018 Revised: 01 January 2019
Primary: 35R02, 49N70; Secondary: 91A13

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.
- Mean field games,
- networks,
- ergodic problems,
- Kirchhoff conditions
Citation: Yves Achdou, Manh-Khang Dao, Olivier Ley, Nicoletta Tchou. A class of infinite horizon mean field games on networks[J]. Networks and Heterogeneous Media, 2019, 14(3): 537-566. doi: 10.3934/nhm.2019021

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Abstract

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.

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