Estimation for stochastic damping hamiltonian systems under partial observation-I. Invariant density

In this paper, we study the non-parametric estimation of the invariant density of some ergodic hamiltonian systems, using kernel estimators. The main result is a central limit theorem for such estimators under partial observation (only the positions are observed). The main tools are mixing estimates and refined covariance inequalities, the main difficulty being the strong degeneracy of such processes. This is the first paper of a series of at least two, devoted to the estimation of the characteristics of such processes: invariant density, drift term, volatility.