Stationary Mean Field Games systems defined on networks

TL;DR

This paper studies stationary Mean Field Games systems on networks, focusing on vertex transition conditions derived from optimal control, and establishes well-posedness, existence, and uniqueness of solutions.

Contribution

It introduces a novel framework for MFG systems on networks with control-based vertex conditions and proves key mathematical properties.

Findings

Well-posedness of the individual equations

Existence of solutions for the MFG system

Uniqueness of the solution

Abstract

We consider a stationary Mean Field Games system defined on a network. In this framework, the transition conditions at the vertices play a crucial role: the ones here considered are based on the optimal control interpretation of the problem. We prove separately the well-posedness for each of the two equations composing the system. Finally, we prove existence and uniqueness of the solution of the Mean Field Games system.