On efficient estimation of invariant density for ergodic diffusion processes

The problem of nonparametric invariant density function estimation of an ergodic diffusion process is considered. The local asymptotic minimax lower bound on the risk of all the estimators is established. The asymptotic risk considered measures the distance between the estimators and the density that has to be estimate in a functional space endowed with the supremum norm. The local time estimator is asymptotically efficient in the sense of this lower bound.