  We study the long time statistics of a two-dimensional Hamiltonian system in
the presence of Gaussian white noise. While the original dynamics is known to
exhibit finite time explosion, we demonstrate that under the impact of the
stochastic forcing as well as a deterministic perturbation, the solutions are
exponentially attractive toward the unique invariant probability measure. This
extends previously established results in which the system is shown to be
noise-induced stable in the sense that the solutions are bounded in
probability.
