A general characterization of the mean field limit for stochastic   differential games
  The mean field limit of large-population symmetric stochastic differential games is derived in a general setting, with and without common noise, on a finite time horizon. Minimal assumptions are imposed on equilibrium strategies, which may be asymmetric and based on full information. It is shown that approximate Nash equilibria in the $n$-player games admit certain weak limits as $n$ tends to infinity, and every limit is a weak solution of the mean field game (MFG). Conversely, every weak MFG solution can be obtained as the limit of a sequence of approximate Nash equilibria in the $n$-player games. Thus, the MFG precisely characterizes the possible limiting equilibrium behavior of the $n$-player games. Even in the setting without common noise, the empirical state distributions may admit stochastic limits which cannot be described by the usual notion of MFG solution. 
Daniel Lacker
math.PR