Georgia Institute of Technology College of Sciences

The study of differential equations on infinite spaces of probability measures has become an active field of research in recent years. Such equations arise when describing nonlinear effects in systems with a large number of interacting particles or agents. A rigorous well-posedness theory for such equations then leads to results about large deviations for interacting particle systems, limiting statements about free energies in mean field spin glasses, and mean field descriptions in high-dimensional optimization and game theory, among many other examples.

 

This talk will give an overview of some recent well-posedness results for Hamilton-Jacobi equations on probablity spaces. The nonlinear and infinite-dimensional nature of the underlying space necessitates a mix of techniques from functional analysis and probability theory. We also discuss how these PDE techniques can be used to deduce qualitative and quantitative convergence results for stochastic control and differential games for a large system of interacting agents. This talk is based on joint work with Joe Jackson (University of Chicago) and Samuel Daudin (Université Paris Cité, Laboratoire Jacques Louis Lions).