2015/03/16 by P. Jameson Graber, Graber, Philip Jameson
Economics, Econometrics and Finance · Mathematics · #35K61 #Analysis of PDEs (math.AP) #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Nonlinear Partial Differential Equations #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1503.04733
openalex publication_date 2015/03/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the short-time existence and uniqueness of solutions to a coupled system of partial differential equations arising in mean field game theory. It has the generic form \ -∂t u - Δu + H(t,x,m,∇ u) = f(t,x,m)
∂t m - Δm - div (m∇p H(t,x,m,∇ u)) = 0. plus initial-final and boundary conditions. The novelty of the problem is that the Hamiltonian H(t,x,m,p) may take such forms as m-α|p|r for some α≥ 0 and r > 1. Our main result is the existence of weak solutions for small times T so long as r is not too large, and uniqueness under additional constraints. The main ingredient in the proof is an a priori estimate on solutions to the Fokker-Planck equation. We also briefly consider existence and uniqueness of solutions to an optimal control problem related to mean field games.