A class of infinite horizon mean field games on networks

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

doi:10.3934/nhm.2019021

Article Contents

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

This issue Previous Article Compressible and viscous two-phase flow in porous media based on mixture theory formulation Next Article Saddle solutions for a class of systems of periodic and reversible semilinear elliptic equations

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
^* Corresponding author: Yves Achdou

Received: July 2018

Revised: January 2019

Published: September 2019

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

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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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Mathematics Subject Classification: Primary: 35R02, 49N70; Secondary: 91A13.

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Figure 1. 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 $

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