2023/03/25 by Wenpin Tang, Yuming Paul Zhang, Tang, Wenpin +1 · 2 citations
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #35K10 #35Q89 #49L25 #91A15 #Advanced Thermodynamics and Statistical Mechanics #Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC) #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2303.14560
openalex publication_date 2023/03/25 · openalex created_date 2023/03/31 · openalex updated_date 2026/07/28
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.