An overlook on statistical inference issues for stochastic damping hamiltonian systems under the fluctuation-dissipation condition

In this paper, we are interested in nonparametric inference issues for stochastic damping hamiltonian systems under the fluctuation-dissipation condition. This condition relates the magnitude of the dissipative term and the magnitude of the random term. The precise balance between the drift term which removes energy in average and the stochastic term provided by the fluctuation-dissipation relation insures that the canonical measure is preserved by the dynamics. In this framework, it is possible to give an explicit construction of a Lyapunov function and thus to prove exponential ergodicity. Then, we consider various estimation procedures and provide also a numerical section, where simulations are conducted.