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

Area of Science:

• Mathematical Physics
• Statistical Mechanics
• Probability Theory

Background:

• Mean field theory describes large interacting particle systems.
• Lévy noise introduces non-Gaussian fluctuations.
• The small mass limit is crucial for modeling particle behavior.

Purpose of the Study:

• To investigate the small mass limit in mean field theory for systems with non-Gaussian Lévy noise.
• To derive limit equations and analyze convergence rates based on the properties of Lévy noise.
• To develop an effective limit model for interacting particles under Lévy fluctuations.

Main Methods:

• Analysis of the small mass limit (ε→0) and mean field limit (N→∞).
• Comparison of different limit orders (mean field then small mass, and vice versa).
• Mathematical derivation of limit equations and rigorous error estimates.

Main Results:

• For finite second moment Lévy noise, limit equations are obtained with rates ε+1/εN or ε+1/N depending on limit order.
• For α-stable Lévy noise (infinite second moment), the small mass limit must precede the mean field limit.
• Specific convergence rates are derived for α-stable noise, dependent on parameters p and α.

Conclusions:

• The order of limits significantly impacts convergence rates for interacting particle systems with Lévy noise.
• A unified limit model is established for systems with non-Gaussian Lévy fluctuations.
• Rigorous error estimates provide a strong foundation for the derived limit equations.