The problem of determining a periodic Lipschitz vector field b=(b1,…,bd) from an observed trajectory of the solution (Xt:0≤t≤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 Wt 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 L2-loss in any dimension, and also for supremum norm loss
when d≤4. Further, when d≤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 μb. The limiting Gaussian process distributions have a covariance
structure that is asymptotically optimal from an information-theoretic point of
view.Comment: 55 pages, to appear in the Annals of Statistic