Twenty years ago or so, Roya Mohayaee, Yann Brenier, Uriel Frisch and some co-authors designed an algorithm based on optimal transport for reconstructing the early Universe. Meanwhile, Yann Brenier proposed a modified theory of gravitation also based on optimal transport (Monge-Ampère gravitation) and built a system of finitely many independent Brownian particles as a bridge between the early and today discretized Universes, once some temperature parameter tends down to zero.

It turns out that Brenier's model can be recasted into a Schrödinger problem based on the limit of the empirical measure of Brownian particles experiencing a mean field quantum interaction, when the number of particles tends to infinity and both the temperature (of the order of Boltzmann constant) and the quantum coupling (of order of Planck's constant) tend down to zero. This a joint work with Roya Mohayaee (Institut d'astrophysique de Paris).