We study a Bayesian approach to nonparametric estimation of the periodic
drift function of a one-dimensional diffusion from continuous-time data. We
rewrite the likelihood in terms of Riemann integrals, by introducing the local
time of the process, and specify a centered Gaussian prior on the drift with a
precision operator that is of differential form. It is proved that this is a
conjugate prior for the likelihood and hence that the posterior is also
Gaussian. We give an explicit expression for the posterior precision operator,
also of differential form, and show that the posterior mean is the solution of
a differential equation requiring inversion of the posterior precision for its
solution. Moreover, we bound the rate at which the posterior contracts around
the true drift function. Our formulation of the estimation problem leads to
algorithms which are readily implementable and analyzed using ideas from the
numerical analysis of differential equations. The central results proved here
require tools from the analysis of differential equations, together with new
functional limit theorems for the local time of diffusions on the circle.