Problems in Mean Field Games and Mean Field Control

This thesis studies various problems that arise in the study of Mean Field Games and Mean Field Control. First we prove that a hypoelliptic MFG system with local-coupling is well posed, extending results from the the parabolic literature. Next, we show that a first-order local MFG system, as well as the planning problem, admit smooth solutions and characterize their long time behavior in one dimension. Finally, we show that for smooth, but not necessarily convex data, there exists a open and dense region, in which the value function in Mean Field Control, as well as the optimal controls, converge at a rate of $\frac{1}{N}$. .