  We consider the question of estimating the drift and the invariant density
for a large class of scalar ergodic diffusion processes, based on continuous
observations, in $\sup$-norm loss. The unknown drift $b$ is supposed to belong
to a nonparametric class of smooth functions of unknown order. We suggest an
adaptive approach which allows to construct drift estimators attaining minimax
optimal $\sup$-norm rates of convergence. In addition, we prove a Donsker
theorem for the classical kernel estimator of the invariant density and
establish its semiparametric efficiency. Finally, we combine both results and
propose a fully data-driven bandwidth selection procedure which simultaneously
yields both a rate-optimal drift estimator and an asymptotically efficient
estimator of the invariant density of the diffusion. Crucial tool for our
investigation are uniform exponential inequalities for empirical processes of
diffusions.
