  We derive uniform concentration inequalities for continuous-time analogues of
empirical processes and related stochastic integrals of scalar ergodic
diffusion processes. Thereby, we lay the foundation typically required for the
study of sup-norm properties of estimation procedures for a large class of
diffusion processes. In the classical i.i.d. context, a key device for the
statistical sup-norm analysis is provided by Talagrand-type concentration
inequalities. Aiming for a parallel substitute in the diffusion framework, we
present a systematic, self-contained approach to such uniform concentration
inequalities via martingale approximation and moment bounds obtained by the
generic chaining method. The developed machinery is of independent
probabilistic interest and can serve as a starting point for investigations of
other processes such as more general Markov processes, in particular
multivariate or discretely observed diffusions. As a first concrete statistical
application, we analyse the sup-norm error of estimating the invariant density
of an ergodic diffusion via the natural local time estimator and the classical
nonparametric kernel density estimator, respectively.
