  We consider nonparametric Bayesian inference in a multidimensional diffusion
model with reflecting boundary conditions based on discrete high-frequency
observations. We prove a general posterior contraction rate theorem in
$L^2$-loss, which is applied to Gaussian priors. The resulting posteriors, as
well as their posterior means, are shown to converge to the ground truth at the
minimax optimal rate over H\"older smoothness classes in any dimension. Of
independent interest and as part of our proofs, we show that certain
frequentist penalized least squares estimators are also minimax optimal.
