Regularity for second order stationary mean-field games

TL;DR

This paper establishes the existence of classical solutions for second order stationary mean-field game systems, expanding the understanding of their regularity and applicability in various control and long-term behavior contexts.

Contribution

It introduces new techniques to prove regularity for a broader class of Hamiltonians and mean-field couplings, including polynomial and logarithmic types.

Findings

Existence of classical solutions for a wide range of mean-field game systems.

Development of iterative and integral methods for a priori estimates.

Extension of previous results to more general Hamiltonians and couplings.

Abstract

In this paper, we prove the existence of classical solutions for second order stationary mean-field game systems. These arise in ergodic (mean-field) optimal control, convex degenerate problems in calculus of variations, and in the study of long-time behavior of time-dependent mean-field games. Our argument is based on the interplay between the regularity of solutions of the Hamilton-Jacobi equation in terms of the solutions of the Fokker-Planck equation and vice-versa. Because we consider different classes of couplings, distinct techniques are used to obtain a priori estimates for the density. In the case of polynomial couplings, we recur to an iterative method. An integral method builds upon the properties of the logarithmic function in the setting of logarithmic nonlinearities. This work extends substantially previous results by allowing for more general classes of Hamiltonians and mean-field assumptions.