  This work is to investigate the (top) Lyapunov exponent for a class of
Hamiltonian systems under small non-Gaussian L\'evy noise. In a suitable moving
frame, the linearisation of such a system can be regarded as a small
perturbation of a nilpotent linear system. The Lyapunov exponent is then
estimated by taking a Pinsky-Wihstutz transformation and applying the
Khas'minskii formula, under appropriate assumptions on smoothness, ergodicity
and integrability. Finally, two examples are present to illustrate our results.
The results characterize the growth or decay rates of a class of dynamical
systems under the interaction between Hamiltonian structures and non-Gaussian
uncertainties.
