  We consider the problem of nonparametric estimation of the drift of a
continuously observed one-dimensional diffusion with periodic drift. Motivated
by computational considerations, van der Meulen e.a. (2014) defined a prior on
the drift as a randomly truncated and randomly scaled Faber-Schauder series
expansion with Gaussian coefficients. We study the behaviour of the posterior
obtained from the prior from a frequentist asymptotic point of view. If the
true data generating drift is smooth, it is proved that the posterior is
adaptive with posterior contraction rates for the $L_2$-norm that are optimal
up to a log factor. Moreover, contraction rates in $L_p$-norms with $p\in
(2,\infty]$ are derived as well.
