  The problem of determining a periodic Lipschitz vector field $b=(b_1, \dots,
b_d)$ from an observed trajectory of the solution $(X_t: 0 \le t \le T)$ of the
multi-dimensional stochastic differential equation \begin{equation*} dX_t =
b(X_t)dt + dW_t, \quad t \geq 0, \end{equation*} where $W_t$ is a standard
$d$-dimensional Brownian motion, is considered. Convergence rates of a
penalised least squares estimator, which equals the maximum a posteriori (MAP)
estimate corresponding to a high-dimensional Gaussian product prior, are
derived. These results are deduced from corresponding contraction rates for the
associated posterior distributions. The rates obtained are optimal up to
log-factors in $L^2$-loss in any dimension, and also for supremum norm loss
when $d \le 4$. Further, when $d \le 3$, nonparametric Bernstein-von Mises
theorems are proved for the posterior distributions of $b$. From this we deduce
functional central limit theorems for the implied estimators of the invariant
measure $\mu_b$. The limiting Gaussian process distributions have a covariance
structure that is asymptotically optimal from an information-theoretic point of
view.
