  We study nonparametric Bayesian models for reversible multi-dimensional
diffusions with periodic drift. For continuous observation paths, reversibility
is exploited to prove a general posterior contraction rate theorem for the
drift gradient vector field under approximation-theoretic conditions on the
induced prior for the invariant measure. The general theorem is applied to
Gaussian priors and $p$-exponential priors, which are shown to converge to the
truth at the minimax optimal rate over Sobolev smoothness classes in any
dimension.
