# LPTMS PhD Proposal: Mean field games

### Responsable: Denis ULLMO + 33 (0)1 69 15 74 76

Mean field games present a new area of research at the boundary between applied mathematics, social sciences, engineering sciences and physics. It has been initiated a decade ago by Pierre-Louis Lions (recipient of the 94 Fields medal) and Jean-Michel Lasry as a new and promising tool to study many problem of social sciences, and with an explicit mention of the influence of concepts coming from physics (the notion of “mean field approximation”). This field has since then grown significantly, and after a period where mainly stylized models where introduced, we witness now the appearance of (necessarily more involved) mean field game models closer to practical applications in finance, vaccination policies, or energy management through smart electronics.

Up to now, the development of Mean Field Games has mainly originated from the mathematics and economic communities. Mean Field Games theory is, however, by essence a multi-disciplinary field for which the input of physicists is much needed. Indeed, as important as they are, the studies of internal consistency and the numerical schemes developed by mathematicians cannot replace the deeper

understanding of the behavior of these models, obtained in particular through powerful approximation schemes, that physicists (and essentially only them) know how to provide.

For physicists a good “entry point” to the problematic of Mean Field Games is through the formal, but deep, connection between Mean Field Games and the nonlinear Schroedinger (or Gross-Pitaevskii) equation. This connection makes it possible to import to the field of Mean Field Games a variety of tools (ranging from exact methods and approximation schemes to intuitive qualitative descriptions) which have been developed along the year by physicists when studying interacting bosons or gravity waves in inviscid fluids.

The general subject of the proposed thesis is the study of Mean Field Games from a physicist point of view, that is with an objective to provide a true understanding (through the identification of the relevant parameters and scale and the development of approximation schemes in the regimes of interest) of the solutions of Mean Field Games equations. More specifically, two possible directions the proposed PhD could take would be:

1. The study of phase transition in Mean Filed games.

2. To use the knowledge obtained on simple models to study more complicated Mean Field Games, and in particular address more realistic (less stylized) Mean Field Games.

These studies should imply a mix between analytical and numerical works, somewhat more shifted on the analytical side.