The convergence rate of vanishing viscosity approximations for mean field games

TL;DR

This paper establishes the first convergence rate results for vanishing viscosity approximations in mean field games, highlighting the impact of dimension, coupling type, and Hamiltonian growth on the rate.

Contribution

It provides novel convergence rate estimates for MFGs with local and nonlocal couplings, revealing potential phase transitions and improving understanding of viscosity limits.

Findings

Convergence rates depend on dimension and coupling type.

Faster rates are achieved for nonlocal, regularizing couplings.

Potential phase transition in dimension dependence based on Hamiltonian growth.

Abstract

Motivated by numerical challenges in first-order mean field games (MFGs) and the weak noise theory for the Kardar-Parisi-Zhang equation, we consider the problem of vanishing viscosity approximations for MFGs. We provide the first results on the convergence rate to the vanishing viscosity limit in mean field games, with a focus on the dimension dependence of the rate exponent. Two cases are studied: MFGs with a local coupling and those with a nonlocal, regularizing coupling. In the former case, we use a duality approach and our results suggest that there may be a phase transition in the dimension dependence of vanishing viscosity approximations in terms of the growth of the Hamiltonian and the local coupling. In the latter case, we rely on the regularity analysis of the solution, and derive a faster rate compared to MFGs with a local coupling. A list of open problems are presented.