  In this paper we consider the nonparametric functional estimation of the
drift of Gaussian processes using Paley-Wiener and Karhunen-Lo\`eve expansions.
We construct efficient estimators for the drift of such processes, and prove
their minimaxity using Bayes estimators. We also construct superefficient
estimators of Stein type for such drifts using the Malliavin integration by
parts formula and stochastic analysis on Gaussian space, in which superharmonic
functionals of the process paths play a particular role. Our results are
illustrated by numerical simulations and extend the construction of James-Stein
type estimators for Gaussian processes by Berger and Wolper.
