On weak solutions to first-order discount mean field games

TL;DR

This paper proves the existence, uniqueness, and stability of weak solutions to first-order discount mean field games, illustrating solution multiplicity and convergence properties with examples.

Contribution

It introduces new existence and uniqueness results for weak solutions and explores solution multiplicity and convergence in mean field games.

Findings

Existence and uniqueness of weak solutions established.

Example of multiple solutions to the ergodic problem provided.

Convergence of weak solutions demonstrated under certain conditions.

Abstract

In this paper, we establish the existence and uniqueness of weak solutions to first-order discount mean field games and a stability result to give the existence for the ergodic problem. We show an example to illustrate the multiplicity of weak solutions to the ergodic problem. With this motivation, we address a selection condition, which is a necessary condition that any limit of solutions under subsequence satisfies. As an application, we show a nontrivial example to get the convergence of weak solutions.