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<α<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−ρ), ρ>α/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.
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BibTeX
@misc{autran2026conditional,
title = {{Conditional Fluctuations of Mean-Field Systems with Stable Common Noise}},
author = {Alexandre Autran},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01391},
eprint = {2610.01391},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01391},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Conditional Fluctuations of Mean-Field Systems with Stable Common Noise
AU - Alexandre Autran
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.01391
UR - https://arxiv.org/abs/2610.01391
AB - We study conditional fluctuations of smooth mean-field systems on a torus with endogenous symmetric stable common jumps of index 0<α<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^−ρ), ρ>α/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.
ER -