Caltech/UCLA Joint Analysis Seminar

The theory of Mean Field Games (MFG) was introduced simultaneously roughly a decade ago by J.-M. Lasry and P.-L. Lions on the one hand and P. Caines, M. Huang and R. Malhamé on the other hand. The aim of both groups was to study limits of Nash equilibria of differential games when the number of players tends to infinity. A fundamental object — introduced by Lions in his lectures — that fully characterizes these limit equilibria is the so-called master equation. This is an infinite dimensional Hamilton-Jacobi equation on the Wasserstein space (of Borel probability measures endowed with a distance arising in the Monge-Kantorovich optimal transport problem). A central question in the theory of MFG is the well-posedness of this equation in various settings. In this talk, we will focus on the first order equation, i.e. without any noise in the dynamics of the agents. Because of the lack of a smoothing effect, only a short time existence result of classical solutions (due to W. Gangbo and A. Swiech) is available. The highly nonlocal nature of the equation together with the non-flat geometry of the Wasserstein space prevent us from developing a theory of viscosity solutions in this setting. After an overview of the subject, in the second half of the talk — as part of an ongoing joint work with W. Gangbo — we will present some connections to the weak KAM theory on the Wasserstein space (developed recently by W. Gangbo and A. Tudorascu).