  The effect of weak multiplicative colored noise on the dynamics of a
Hamiltonian system is studied by means of asymptotic methods, in the vicinity
of homoclinic or heteroclinic trajectories. A general expression for the
probability of noise-induced separatrix crossing is obtained, and is
illustrated by means of a two-well Duffing oscillator. It shows how weak noise
can significantly affect the dynamics near separatrices. In addition, the
influence of the degree of non-linearity of the noise amplitude is discussed.
