  As a starting point we prove a functional central limit theorem for
estimators of the invariant measure of a geometrically ergodic Harris-recurrent
Markov chain in a multi-scale space. This allows to construct confidence bands
for the invariant density with optimal (up to undersmoothing)
$L^{\infty}$-diameter by using wavelet projection estimators. In addition our
setting applies to the drift estimation of diffusions observed discretely with
fixed observation distance. We prove a functional central limit theorem for
estimators of the drift function and finally construct adaptive confidence
bands for the drift by using a completely data-driven estimator.
