  The dynamics of a weakly dissipative Hamiltonian system submitted to
stochastic perturbations has been investigated by means of asymptotic methods.
The probability of noise-induced separatrix crossing, which drastically changes
the fate of the system, is derived analytically in the case where noise is an
additive Kubo-Anderson process. This theory shows how the geometry of the
separatrix, as well as the noise intensity and correlation time, affect the
statistics of crossing. Results can be applied to a wide variety of systems,
and are valid in the limit where the noise correlation time scale is much
smaller than the time scale of the undisturbed Hamiltonian dynamics.
