Fluctuation for interacting particle systems with common noise
Paul Nikolaev
TL;DR
The paper analyzes fluctuations in large interacting particle systems on $\\mathbb{R}^d$ subjected to both idiosyncratic and common noise, with a nonlocal interaction kernel $k \in L^2 \cap L^\ fty$. It develops a framework based on uniform relative entropy bounds and Kolmogorov compactness to show tightness and convergence of the fluctuation process $\\eta^N = \sqrt{N}(\\mu_t^N - \\rho_t)$ to the unique solution of a linear SPDE, even in the unbounded domain and with non-Lipschitz interactions. A key challenge is that the limit is Gaussian only conditionally on the common noise, requiring a conditional covariation analysis and a filtration-aware identification of the limit. The results extend previous fluctuation theories beyond Lipschitz kernels and bounded domains, providing a rigorous foundation for Gaussian fluctuations under common noise in mean-field-type models with multiplicative diffusion. This advances the mathematical understanding of fluctuations in systems with aggregated shocks and has implications for mean-field games and financial models with systemic risk.
Abstract
We consider the asymptotic behavior of the fluctuation process for large stochastic systems of interacting particles driven by both idiosyncratic and common noise with an interaction kernel \(k \in L^2(\R^d) \cap L^\infty(\R^d)\). Our analysis relies on uniform relative entropy estimates and Kolmogorov's compactness criterion to establish tightness and convergence of the fluctuation process. In this framework, an extension of the exponential law of large numbers is used to derive the necessary uniform estimates, while a conditional Fubini theorem is employed in the identification of the limit in the presence of common noise. We demonstrate that the fluctuation process converges in distribution to the unique solution of a linear stochastic evolution equation. This work extends previous fluctuation results beyond the classical Lipschitz framework.