Regularity and long time behavior of one-dimensional first-order mean field games and the planning problem

TL;DR

This paper investigates the regularity and long-term behavior of one-dimensional first-order mean field games, establishing new regularity results and characterizing exponential convergence to steady states using displacement convexity methods.

Contribution

It introduces two new regularity results for solutions of the mean field games system without requiring blow-up assumptions, and characterizes their long time behavior with exponential turnpike property.

Findings

Existence of classical solutions without blow-up assumptions.

Interior smoothness of weak solutions without bounded away from zero densities.

Solutions exhibit exponential convergence to the steady state.

Abstract

We study the regularity and long time behavior of the one-dimensional, local, first-order mean field games system and the planning problem, assuming a Hamiltonian of superlinear growth, with a non-separated, strictly monotone dependence on the density. We improve upon the existing literature by obtaining two regularity results. The first is the existence of classical solutions without the need to assume blow-up of the cost function near small densities. The second result is the interior smoothness of weak solutions without the need to assume neither blow-up of the cost function nor that the initial density be bounded away from zero. We also characterize the long time behavior of the solutions, proving that they satisfy the turnpike property with an exponential rate of convergence, and that they converge to the solution of the infinite horizon system. Our approach relies on the elliptic structure of the system and displacement convexity estimates. In particular, we apply displacement convexity methods to obtain both global and local a priori lower bounds on the density.