Conditional Fluctuations of Mean-Field Systems with Stable Common Noise

Abstract

We study conditional fluctuations of smooth mean-field systems on a torus with endogenous symmetric stable common jumps of index $0<\alpha<1$. A positive-strip estimate makes empirical Poisson acceptance errors negligible at the sampling scale. The conditional laws given the full Poisson master converge in probability to a centered Gaussian kernel in a negative Sobolev path space. In dimension one, quantile coupling preserves this limit for microscopic marks with relative tail remainder $O(x^{-\rho})$, $\rho>\alpha/2$. A quantified second-order tail expansion gives the endogenous rank response: it shifts the conditional Gaussian mean at criticality and, when nonzero, dominates sampling below criticality. We also identify a separate stable-to-Brownian transition for the rank error. Finally, in a translation model, rank and clock couplings give the same microscopic laws. The rank coupling has a functional Gaussian limit. The clock coupling has no diverging additive path normalization at the limiting center.