Homogenization of the backward-forward mean-field games systems in periodic environments

TL;DR

This paper investigates the homogenization of viscous backward-forward mean-field games systems in periodic environments, analyzing the small viscosity limit and the effects of different coupling types on the limit system.

Contribution

It provides the first homogenization results for viscous backward-forward mean-field games systems with separated Hamiltonians and different coupling scenarios.

Findings

Limit system is a first-order forward-backward system.

Nonlocal coupling yields a well-posed mfg-type limit.

Local coupling results depend on smoothness and well-prepared data.

Abstract

We study the homogenization properties in the small viscosity limit and in periodic environments of the (viscous) backward-forward mean-field games system. We consider separated Hamiltonians and provide results for systems with (i) "smoothing" coupling and general initial and terminal data, and (ii) with "local coupling" but well-prepared data.The limit is a first-order forward-backward system. In the nonlocal coupling case, the averaged system is of mfg-type, which is well-posed in some cases. For the problems with local coupling, the homogenization result is proved assuming that the formally obtained limit system has smooth solutions with well prepared initial and terminal data. It is also shown, using a very general example (potential mfg), that the limit system is not necessarily of mfg-type.