I will present some regularity results on a variational problem for curves of probability densities where their velocity is penalized in terms of the \( L^1 \) (or total variation) norm, of the time derivative. This is motivated by a new class of Mean Field Games whose goal is to address models for the real estate market, where agents choose a trajectory which is not continuous in time but has jumps. Each jump corresponds to moving from an address to another, and has a fixed cost. In some cases, the equilibrium is variational and it is possible to state an eulerian problem on curves of densities. The aim of the regularity study is both qualitative and mathematical : we want to see whether the solutions evolve smoothly or also have jumps, and prove some integrability results which seem relevant for the MFG analysis of these models.

This is a joint work with Annette Dumas, who starts a PhD in Lyon 1 on this topic.