Sharp adaptive drift estimation for ergodic diffusions: The multivariate case

We consider estimation of the drift function for a large class of multidimensional ergodic diffusions and establish the exact constant of the risk asymptotics in the L2 risk. The constant is of Pinsker-type and in particular reflects the dependence of the drift estimation problem on the geometry of the diffusion coefficient. In addition, an exact data-driven estimation procedure is proposed, attaining the optimal constant under natural L2 Sobolev smoothness conditions on the drift.