Small mass limit for stochastic interacting particle systems with Lvy noise and linear alignment force

We study the small mass limit in mean field theory for an interacting particle system with non-Gaussian Levy noise. When the Levy noise has a finite second moment, we obtain the limit equation with convergence rate epsilon + 1/epsilon N, by taking first the mean field limit N -> infinity and then the small mass limit epsilon -> 0. If the order of the two limits is exchanged, the limit equation remains the same but has a different convergence rate epsilon + 1 / N. However, when the Levy noise is alpha-stable, which has an infinite second moment, we can only obtain the limit equation by taking first the small mass limit and then the mean field limit, with the convergence rate 1 /N alpha-1 + 1/N-p2 + epsilon(p/alpha )where p is an element of (1 , alpha). This provides an effectively limit model for an interacting particle system under a non-Gaussian Levy fluctuation, with rigorous error estimates.