Abstract

The theory of mean field games aims to describe the limits of Nash equilibria for differential games as the number of players tends to infinity. If players control their state by choosing their acceleration, then the mean field games system describing this equilibrium includes a kinetic transport term. Previous results on the well-posedness theory of mean field games of this type assume either that the running and final costs are regularising functionals of the density variable, or the presence of noise – that is, a second-order system. I will present recently obtained results in which we construct global-in-time weak solutions for a deterministic `kinetic’ mean field game with local (hence non-regularising) couplings, under suitable convexity and monotonicity conditions. Our approach is based on a characterisation of the solutions through two optimisation problems in duality. Furthermore, under stronger monotonicity/convexity assumptions, we obtain Sobolev regularity estimates on the solutions. This talk is based on joint work with Alpár Mészáros.