Homogenization of Dissipative Hamiltonian Systems under L\'evy Fluctuations

TL;DR

This paper derives a limiting equation for dissipative Hamiltonian systems influenced by multiplicative Le9vy noise, revealing noise-induced drifts and establishing convergence results with an illustrative example.

Contribution

It introduces a small mass limit for stochastic Hamiltonian systems with Le9vy noise, including convergence proofs and a Le9vy Smoluchowski-Kramers approximation.

Findings

Convergence to the limiting equation in probability.

Convergence in moments under stronger conditions.

A Le9vy Smoluchowski-Kramers approximation example.

Abstract

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.