Mass transfers and Optimal Stochastic Transports

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There are three main thrusts to my research proposal. The first is a relatively new direction focused on stochastic optimal transport problems and the theory of dynamical mean field games with free end-time. The second is an asymptotic theory of mass transfers, an encompassing concept introduced recently by the applicant, while the third is a continuation of my long-term research project on borderline elliptic boundary value problems involving the Hardy-Schrodinger operator. The theory of optimal martingale mass transport originated in mathematical finance and is now well understood in the one-dimensional case, where it models the evolution of the price of only one stock. However, open questions abound in the higher dimensional case, i.e., for "multi-stock" models, where the problems are much more challenging. The case of Brownian martingales is particularly interesting as it connects these topics to problems of optimal stopping of Brownian motion and the classical theory of Skorokhod embeddings. This in turn leads to the relatively new direction of optimal stochastic transportation with free end time, where the optimization is over stopping times and processes with prescribed end distributions. The Eulerian formulation of stochastic mass transport theory is closely connected to the theory of mean field games, which originated independently in economics, engineering, and mathematical analysis. This topic can be loosely described as the study of strategic decision-making in very large populations of small interacting agents. Mathematically, it can be described as a limit of a multi-player Nash equilibrium as the number of players becomes very large. Mean field games with fixed end times have been the subject of intensive investigations in recent years. However, mean field games with free end time have only been considered in a few simple examples and our focus will be on this case which contains new phenomena that have not been captured by previous models. Another related set of problems stems from the general theory of mass transfers and the asymptotic properties of their associated non-linear Kantorovich operators. Natural connections to large deviation theory, as well as applications to symbolic dynamics are being developed. The non-trivial extensions to a non-compact setting will be actively pursued. A theory of multilinear transfers between several probability distributions -as opposed to pairs- is also under development. A long-term research project of the proposer and his collaborators has been concerned with the fine analysis of second order Hardy-Schrodinger operators on domains in Euclidean space, when the singularity (zero) lies either on the boundary of the domains under study or in their interior. The next phase will focus on issues of existence, multiplicity and qualitative properties of solutions of PDEs involving Hardy-Schrodinger operators on hyperbolic manifolds, as well as their fractional counterparts.