Convergence rate of Smoluchowski--Kramers approximation with stable L\'{e}vy noise

The small mass limit of the Langevin equation perturbed by $\alpha$-stable L\'{e}vy noise is considered by rewriting it in the form of slow-fast system, and spliting the fast component into three parts, where $\alpha\in(1,2)$. By exploring the three parts respectively, the approximation equation is derived. The convergence is either in the sense of uniform metric or in the sense of Skorokhod metric, depending on how regular the noise is. In the former case, we obtain the convergence rate.