  The paper studies the convergence, as $N$ tends to infinity, of a system of
$N$ coupled Hamilton-Jacobi equations, the Nash system. This system arises in
differential game theory. We describe the limit problem in terms of the
so-called "master equation", a kind of second order partial differential
equation stated on the space of probability measures. Our first main result is
the well-posedness of the master equation. To do so, we first show the
existence and uniqueness of a solution to the "mean field game system with
common noise", which consists in a coupled system made of a backward stochastic
Hamilton-Jacobi equation and a forward stochastic Kolmogorov equation and which
plays the role of characteristics for the master equation. Our second main
result is the convergence, in average, of the solution of the Nash system and a
propagation of chaos property for the associated "optimal trajectories".
