  Mean field games (MFGs) describe the limit, as $n$ tends to infinity, of
stochastic differential games with $n$ players interacting with one another
through their common empirical distribution. Under suitable smoothness
assumptions that guarantee uniqueness of the MFG equilibrium, a form of law of
large of numbers (LLN), also known as propagation of chaos, has been
established to show that the MFG equilibrium arises as the limit of the
sequence of empirical measures of the $n$-player game Nash equilibria,
including the case when player dynamics are driven by both idiosyncratic and
common sources of noise. The proof of convergence relies on the so-called
master equation for the value function of the MFG, a partial differential
equation on the space of probability measures. In this work, under additional
assumptions, we establish a functional central limit theorem (CLT) that
characterizes the limiting fluctuations around the LLN limit as the unique
solution of a linear stochastic PDE. The key idea is to use the solution to the
master equation to construct an associated McKean-Vlasov interacting
$n$-particle system that is sufficiently close to the Nash equilibrium dynamics
of the $n$-player game for large $n$. We then derive the CLT for the latter
from the CLT for the former. Along the way, we obtain a new multidimensional
CLT for McKean-Vlasov systems. We also illustrate the broader applicability of
our methodology by applying it to establish a CLT for a specific
linear-quadratic example that does not satisfy our main assumptions, and we
explicitly solve the resulting stochastic PDE in this case.
