Singular perturbation for linear SDE with mean-field interaction (Q2703054)

Singularly perturbed time-varying affine differential equations with additive white noise are considered. The drift depends upon a positive parameter \(\gamma\) which represents the intensity of the interaction. For the scalar SDE with mean field of the McKean type, which is tagged to the limit behaviour as \(N\to\infty\) of the interacting \(N\) particle system, it is proved, under some assumptions, that the fast component multiplied by \(\sqrt\varepsilon\) converges in distribution to a normal distribution with zero mean and variance depending on \(\gamma\) as \(\varepsilon\to 0\) and the slow component converges in mean square to a diffusion process with the diffusion coefficient depending on \(\gamma\) as \(\varepsilon\to 0\). Also, the interchangeability of the limits is discussed.