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

We study the small mass limit in mean field theory for an interacting particle system with non-Gaussian Lévy noise. When the Lévy noise has a finite second moment, we obtain the limit equation with convergence rate ε+1/εN, by taking first the mean field limit N→∞ and then the small mass limit ε→0. If the order of the two limits is exchanged, the limit equation remains the same but has a different convergence rate ε+1/N. However, when the Lévy noise is α-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/Nα-1+1/Np2+εp/α where p∈(1,α). This provides an effectively limit model for an interacting particle system under a non-Gaussian Lévy fluctuation, with rigorous error estimates.

Exponential ergodicity for a class of Markov processes with interactions

We establish exponential ergodicity for a class of Markov processes with interactions, including two-factor type processes and Gruschin type processes. The proof is elementary and direct via the Markov coupling technique.