Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions

The problem of determining a periodic Lipschitz vector fieldb=(b1,...,bd) from an observed trajectory of the solution (Xt: 0≤t≤T) of the multi-dimensional stochastic differential equationdXt=b(Xt)dt+dWt, t≥0,whereWtis a standardd-dimensional Brownian motion, is consid-ered. Convergence rates of a penalised least squares estimator, whichequals the maximum a posteriori (MAP) estimate corresponding to ahigh-dimensional Gaussian product prior, are derived. These resultsare deduced from corresponding contraction rates for the associatedposterior distributions. The rates obtained are optimal up to log-factors inL2-loss in any dimension, and also for supremum norm losswhend≤4. Further, whend≤3, nonparametric Bernstein-von Misestheorems are proved for the posterior distributions ofb. From this wededuce functional central limit theorems for the implied estimatorsof the invariant measureμb. The limiting Gaussian process distribu-tions have a covariance structure that is asymptotically optimal froman information-theoretic point of view