Convergence of the solutions of the MFG discounted Hamilton-Jacobi equation

TL;DR

This paper proves the full convergence of solutions to the discounted Hamilton-Jacobi equation in Wasserstein space from potential MFGs to a corrector function, adapting finite-dimensional techniques without mollification.

Contribution

It extends the convergence analysis of discounted Hamilton-Jacobi equations to Wasserstein space, characterizing the limit via smooth Mather measures without mollification.

Findings

Proves full convergence of $\mathcal V_\delta$ to $\chi_0$ in Wasserstein space.

Characterizes the limit $\chi_0$ through smooth Mather measures.

Adapts finite-dimensional proof techniques to the Wasserstein setting.

Abstract

We consider the solution $\mathcal V_\delta$ of the discounted Hamilton-Jacobi equation in the Wasserstein space arising from potential MFG and we prove its full convergence to a corrector function $\chi_0$. We follow the structure of the proof of the analogue result in the finite dimensional setting provided by Davini, Fathi, Iturriaga, Zavidovique in 2017. We characterize the limit $\chi_0$ through a particular set of smooth Mather measures. A major point that distinguishes the techniques deployed in the standard setting from the ones that we use here is the lack of mollification in the Wasserstain space.