Mean Field Games in the weak noise limit : A WKB approach to the Fokker-Planck equation

TL;DR

This paper applies the WKB method to solve the Fokker-Planck equation in the small noise limit for a Mean Field Game model, providing a transparent approach to the time-dependent problem.

Contribution

It introduces a WKB approach to analyze the Fokker-Planck equation in the weak noise limit within a Mean Field Game context, offering a novel analytical solution method.

Findings

WKB method effectively solves the Fokker-Planck equation in the small noise regime.

Provides explicit solutions for the time-dependent diffusion problem.

Enhances understanding of Mean Field Games with weak stochastic effects.

Abstract

Motivated by the study of a Mean Field Game toy model called the "seminar problem", we consider the Fokker-Planck equation in the small noise regime for a specific drift field. This gives us the opportunity to discuss the application to diffusion problem of the WKB approach "a la Maslov", making it possible to solve directly the time dependant problem in an especially transparent way.