  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.
