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

Author:

Wang Zibo1ORCID,Lv Li1ORCID,Zhang Yanjie2ORCID,Duan Jinqiao34ORCID,Wang Wei5ORCID

Affiliation:

1. School of Mathematics and Statistics and Center for Mathematical Sciences, Huazhong University of Science and Technology 1 , Wuhan 430074, China

2. Henan Academy of Big Data, Zhengzhou University 2 , Zhengzhou 450001, China

3. Department of Mathematics and Department of Physics, Great Bay University 3 , Dongguan 523000, China

4. Center for Mathematical Sciences, Huazhong University of Science and Technology 4 , Wuhan 430074, China

5. Department of Mathematics, Nanjing University 5 , Nanjing 210093, China

Abstract

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.

Funder

National Natural Science Foundation of China

Publisher

AIP Publishing

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