Semi-linear parabolic equations on homogeneous Lie groups arising from mean field games

TL;DR

This paper investigates the existence and uniqueness of solutions to certain semilinear parabolic equations on homogeneous Lie groups, addressing anisotropic geometric challenges, and applies these results to establish short-time solutions for mean field games systems.

Contribution

It provides new existence and uniqueness results for semilinear parabolic equations on homogeneous Lie groups, considering anisotropic geometry, and applies these to mean field games.

Findings

Existence and uniqueness of solutions for Fokker-Planck and Hamilton-Jacobi equations.

Analysis of anisotropic geometry effects on PDE solutions.

Short-time existence of classical solutions for mean field games systems.

Abstract

The existence and the uniqueness of solutions to some semilinear parabolic equations on homogeneous Lie groups, namely, the Fokker-Planck equation and the Hamilton-Jacobi equation, are addressed. The anisotropic geometry of the state space plays a crucial role in our analysis and creates several issues that need to be overcome. Indeed, the ellipticity directions span, at any point, subspaces of dimension strictly less than the dimension of the state space. Finally, the above results are used to obtain the short-time existence of classical solutions to the mean field games system defined on an homogenous Lie group.