<p>This work investigates the stochastic dynamics of Hamiltonian systems with hyperbolic structure under external noise. To overcome the conflict between the non-anticipative nature of stochastic solutions and the exponential dichotomies of the hyperbolic structure, we construct auxiliary processes that are distributionally equivalent to the original dynamics. This construction allows us to leverage both explicit stable/unstable splittings (when available) and the Oseledets decomposition provided by the Multiplicative Ergodic Theorem (in the fully stochastic case). Within this framework, we prove central limit theorems and functional central limit theorems for the time-integrated normal deviations, with limiting covariances given explicitly in terms of the system parameters. These results establish the distributional characterization of hyperbolic tori persistence under stochastic perturbations, illustrating how tools from stochastic analysis and ergodic theory yield precise answers to a classical problem in Hamiltonian dynamics.</p>