  We study the dynamics of the large N limit of the Kuramoto model of coupled
phase oscillators, subject to white noise. We introduce the notion of shadow
inertial manifold and we prove their existence for this model, supporting the
fact that the long term dynamics of this model is finite dimensional. Following
this, we prove that the global attractor of this model takes one of two forms.
When coupling strength is below a critical value, the global attractor is a
single equilibrium point corresponding to an incoherent state. Conversely, when
coupling strength is beyond this critical value, the global attractor is a
two-dimensional disk composed of radial trajectories connecting a saddle
equilibrium (the incoherent state) to an invariant closed curve of locally
stable equilibria (partially synchronized state). Our analysis hinges, on the
one hand, upon sharp existence and uniqueness results and their consequence for
the existence of a global attractor, and, on the other hand, on the study of
the dynamics in the vicinity of the incoherent and synchronized equilibria. We
prove in particular non-linear stability of each synchronized equilibrium, and
normal hyperbolicity of the set of such equilibria. We explore mathematically
and numerically several properties of the global attractor, in particular we
discuss the limit of this attractor as noise intensity decreases to zero.
