Non coercive unbounded first order Mean Field Games: the Heisenberg example

TL;DR

This paper investigates first order Mean Field Games on the Heisenberg group, establishing existence of weak solutions and their Lagrangian representation despite non-coercive Hamiltonians and quadratic growth conditions.

Contribution

It introduces a framework for analyzing MFGs in the Heisenberg group with non-coercive Hamiltonians and provides existence and representation results for solutions.

Findings

Existence of weak solutions for the MFG system in the Heisenberg group.

Representation of solutions via Lagrangian formulation.

Generalizations to Heisenberg-type structures.

Abstract

In this paper we study evolutive first order Mean Field Games in the Heisenberg group; each agent can move in the whole space but it has to follow "horizontal" trajectories which are given in terms of the vector fields generating the group and the kinetic part of the cost depends only on the horizontal velocity. The Hamiltonian is not coercive in the gradient term and the coefficients of the first order term in the continuity equation may have a quadratic growth at infinity. The main results of this paper are two: the former is to establish the existence of a weak solution to the Mean Field Game systems while the latter is to represent this solution following the Lagrangian formulation of the Mean Field Games. We also provide some generalizations to Heisenberg-type structures.