  It has recently been shown that there are substantial differences in the
regularity behavior of the empirical process based on scalar diffusions as
compared to the classical empirical process, due to the existence of diffusion
local time. Besides establishing strong parallels to classical theory such as
Ossiander's bracketing CLT and the general Gin\'e-Zinn CLT for uniformly
bounded families of functions, we find increased regularity also for
multivariate ergodic diffusions, assuming that the invariant measure is finite
with Lebesgue density $\pi$. The effect is diminishing for growing dimension
but always present. The fine differences to the classical iid setting are
worked out using exponential inequalities for martingales and additive
functionals of continuous Markov processes as well as the characterization of
the sample path behavior of Gaussian processes by means of the generic chaining
bound. To uncover the phenomenon, we study a smoothed version of the empirical
diffusion process. It turns out that uniform weak convergence of the smoothed
empirical diffusion process under necessary and sufficient conditions can take
place with even exponentially small bandwidth in dimension $d=2$, and with
strongly undersmoothing bandwidth choice for parameters $\beta > d/2$ in case
$d\geq 3$, assuming that the coordinates of drift and diffusion coefficient
belong to some H\"older ball with parameter $\beta$.
