The continuity equation in the Heisenberg-periodic case: a representation formula and an application to Mean Field Games

TL;DR

This paper develops a representation formula for the continuity equation on the Heisenberg group with periodic data and applies it to establish the equivalence of weak and mild solutions in first order Mean Field Games on this group.

Contribution

It introduces a novel representation of solutions to the continuity equation on the Heisenberg group and applies it to analyze Mean Field Games in this non-commutative setting.

Findings

Representation formula for the continuity equation on $\

Weak solutions are also mild solutions in the Heisenberg group setting.

Abstract

We provide a representation of the weak solution of the continuity equation on the Heisenberg group $\mathbb H^1$ with periodic data (the periodicity is suitably adapted to the group law). This solution is the push forward of a measure concentrated on the flux associated with the drift of the continuity equation. Furthermore, we shall use this interpretation for proving that weak solutions to first order Mean Field Games on $\mathbb H^1$ are also mild solutions.