Random dynamics for N SPDEs with mean-field interaction

We investigate the random dynamics of a system consisting of N interacting stochastic partial differential equations (SPDEs) with a mean-field interaction, where the interacting potential is Lipschitz continuous and odd. Leveraging the mean-field structure, we decompose the system into its ensemble average and the fluctuation component. A Lyapunov–Perron method is then employed to establish the existence of a finite-dimensional random invariant manifold for large interaction. Further we give a mean-field limit approximation of the reduced system on the random invariant manifold. Our result shows that random dynamics of the N interacting particle system is determined by that of the ensemble average part of the system for large interaction and the approximating system is deterministic as N → ∞. At last our results are clearly illustrated by one example.