Homogenization of Dissipative Hamiltonian Systems Under Levy Fluctuations

We study the small mass limit for a class of Hamiltonian systems with multiplicative non-Gaussian Levy noise. Derivation of the limiting equation depends on the structure of the stochastic Hamiltonian systems, in which a discontinuous noise-induced drift term arises. Firstly, we show that the momentum in the stochastic Hamiltonian system converges to zero when the kinetic energy has polynomial growth. Then, we prove that the stochastic Hamiltonian system with classical kinetic energy converges to the limiting equation in probability, with respect to Skorokhod topology as the mass tends to zero.