  We study a Bayesian approach to nonparametric estimation of the periodic
drift function of a one-dimensional diffusion from continuous-time data.
Rewriting the likelihood in terms of local time of the process, and specifying
a Gaussian prior with precision operator of differential form, we show that the
posterior is also Gaussian with precision operator also of differential form.
The resulting expressions are explicit and lead to algorithms which are readily
implementable. Using new functional limit theorems for the local time of
diffusions on the circle, we bound the rate at which the posterior contracts
around the true drift function.
