Mean field limits of interacting particle systems with positive stable jumps

TL;DR

This paper investigates the mean field limits of interacting particle systems with positive stable jumps, improving error bounds by using a specialized distance that accounts for large jumps in the system.

Contribution

It introduces a new distance metric after a space transformation to better handle large jumps, enhancing convergence bounds for systems with positive stable jumps.

Findings

Improved error bounds for mean field convergence with positive stable jumps.

A new distance metric that accounts for large jumps in the system.

Application of a concave space transform to trajectories for better analysis.

Abstract

This note is a companion article to the recent paper L\"ocherbach, Loukianova, Marini (2024). We consider mean field systems of interacting particles. Each particle jumps with a jump rate depending on its position. When jumping, a macroscopic quantity is added to its own position. Moreover, simultaneously, all other particles of the system receive a small random kick which is distributed according to a positive $\alpha-$stable law and scaled in $N^{-1/\alpha},$ where $0 < \alpha < 1.$ In between successive jumps of the system, the particles follow a deterministic flow with drift depending on their position and on the empirical measure of the total system. In a more general framework where jumps and state space do not need to be positive, we have shown in L\"ocherbach, Loukianova, Marini (2024) that the mean field limit of this system is a McKean-Vlasov type process which is solution of a non-linear SDE, driven by an $ \alpha-$stable process. Moreover we have obtained in L\"ocherbach, Loukianova, Marini (2024) an upper bound for the strong rate of convergence with respect to some specific distance disregarding big jumps of the limit stable process. In the present note we consider the specific situation where all jumps are positive and particles take values in $[ 0, + \infty [ . $ We show that in this case it is possible to improve upon the error bounds obtained in L\"ocherbach, Loukianova, Marini (2024) by using an adhoc distance obtained after applying a concave space transform to the trajectories. The distance we propose here takes into account the big jumps of the limit $ \alpha-$stable subordinator.