Homogenization of Dissipative Hamiltonian Systems under L\'evy Fluctuations

This work is devoted to deriving small mass limiting equation for a class of Hamiltonian systems with multiplicative L'evy noise. Derivation of the limiting equation depends on the structure of the stochastic Hamiltonian systems, in which a noise-induced drift term arises. We prove convergence to the limiting equation in probability under appropriate assumptions on smoothness and boundedness. Furthermore, we demonstrate convergence in moment under stronger assumptions. A L'evy type Smoluchowski-Kramers approximation result is presented as an illustrative example.