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

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.