  We study the empirical process arising from a multi-dimensional diffusion
process with periodic drift and diffusivity. The smoothing properties of the
generator of the diffusion are exploited to prove the Donsker property for
certain classes of smooth functions. We partially generalise the finding from
the one-dimensional case studied in [van der Vaart & van Zanten, 2005]: that
the diffusion empirical process exhibits stronger regularity than in the
classical case of i.i.d. observations. As an application, precise asymptotics
are deduced for the Wasserstein-1 distance between the time-$T$ occupation
measure and the invariant measure in dimensions $d\leq3$.
