A Class of Infinite Horizon Mean Field Games on Networks

TL;DR

This paper studies infinite horizon mean field games on networks, establishing existence and uniqueness of solutions involving coupled Hamilton-Jacobi-Bellman and Fokker-Planck equations with complex boundary conditions.

Contribution

It introduces a novel framework for mean field games on networks, including the analysis of boundary conditions and proving existence and uniqueness of solutions.

Findings

Existence of solutions under certain assumptions

Uniqueness of solutions for the coupled system

Characterization of boundary and transmission conditions

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.