On the conservative character of discretizations to Itô-Hamiltonian systems with small noise

In this paper, we consider stochastic Hamiltonian systems of Itô type driven by a multiplicative small noise. It is well-known, indeed, that stochastic Hamiltonian problems are suitable candidates to reconcile the Hamiltonian classical mechanics with the non-differentiability of the Wiener process, which provides the innovative character of stochastic effects. In particular, in this work, we provide a characterization of the behavior of averaged Hamiltonians arised in such systems, with more emphasis on the separable and quadratic Hamiltonians. Next, we show that, in general, first order approximations to such systems are not able to retain the same behavior discovered for the exact averaged Hamiltonian. Hence, the analysis for the specific case of ϑ-methods is performed. Finally, numerical evidence is depicted in order to confirm theoretical results.