  We consider inference in the scalar diffusion model
$dX_t=b(X_t)dt+\sigma(X_t)dW_t$ with discrete data $(X_{j\Delta_n})_{0\leq j
\leq n}$, $n\to \infty,~\Delta_n\to 0$ and periodic coefficients. For $\sigma$
given, we prove a general theorem detailing conditions under which Bayesian
posteriors will contract in $L^2$-distance around the true drift function $b_0$
at the frequentist minimax rate (up to logarithmic factors) over Besov
smoothness classes. We exhibit natural nonparametric priors which satisfy our
conditions. Our results show that the Bayesian method adapts both to an unknown
sampling regime and to unknown smoothness.
