Partial Differential Equation Methods for Mean Field Games

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Mean field games are models for problems in which a large number of individuals need to make decisions which are related to those of many other individuals. Many applications of mean field games arise from economics. One example to be treated is modeling the decision of households to allocate their income between savings and consumption; the benefits of a household's decision depend on the decisions that the other households make, since the interest rate in this example is determined through the aggregation of the decisions of all the households. This research will seek to understand mathematical theory of these models; this includes proving that the models have solutions and understanding how these solutions depend upon parameters present in the models. Developing the mean field games theory can have impact on the quality of economic forecasts and economic decision-making. Several graduate and undergraduate students will be trained through participation in this research projectExistence and regularity theory will be developed for solutions of mean field games models, and important asymptotic problems will be investigated. The regularity theory includes demonstrating analytic and Gevrey regularity for solutions which have previously been proved to exist with finite Sobolev regularity. Asymptotic regimes include rigorously studying the limit of differential games with finitely many players as the number of players tends to infinity, studying the limit as diffusion parameters vanish, and studying the limit as the time horizon goes to infinity. Problems with nonseparable Hamiltonian and problems arising from applications, such as the household savings problem, will be emphasized. Mean field games systems are taken with initial and terminal data rather than just initial data; tools from initial value problems in areas such as fluid dynamics will be adapted to the forward-backward setting of mean field games.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.