Estimating the characteristics of stochastic damping Hamiltonian systems from continuous observations

We consider nonparametric invariant density and drift estimation for a class of multidimensional degenerate resp. hypoelliptic diffusion processes, so-called stochastic damping Hamiltonian systems or kinetic diffusions, under anisotropic smoothness assumptions on the unknown functions. The analysis is based on continuous observations of the process, and the estimators' performance is measured in terms of the sup-norm loss. Regarding invariant density estimation, we obtain highly nonclassical results for the rate of convergence, which reflect the inhomogeneous variance structure of the process. Concerning estimation of the drift vector, we suggest both non-adaptive and fully data-driven procedures. All of the aforementioned results strongly rely on tight uniform moment bounds for empirical processes associated to deterministic and stochastic integrals of the investigated process, which are also proven in this paper.