A Logarithmic Fluctuation Hierarchy for Sequential Interacting Diffusions

Abstract

We study Gaussian fluctuations for a lower-triangular system of interacting diffusions in which particle $i$ interacts only with its predecessors. Although the empirical measure of this system converges to the same McKean--Vlasov limit as in the corresponding exchangeable mean-field system, the sequential structure remains visible at the $N^{-1/2}$ fluctuation scale. We introduce the logarithmically weighted fluctuation fields \[ Y_t^{N,n} = \frac{1}{\sqrt{N}} \sum_{i=1}^N \frac{\bigl(\log(N/i)\bigr)^n}{n!} (\delta_{X_t^i}-\bar\rho_t), \qquad n\ge 0, \] and prove joint convergence of the entire family in a countable product of weighted negative Sobolev path spaces to the unique probabilistically strong solution of a linear hierarchy in which $Y^n$ couples to $Y^{n+1}$. In particular, the limit of the empirical fluctuation field $\sqrt{N}(\mu_t^N-\bar\rho_t)=Y_t^{N,0}$ is not governed by the closed fluctuation SPDE arising in the classical exchangeable case. The proof combines conditional-measure replacement, deterministic estimates for the logarithmic weights, a martingale argument, and a weighted Volterra estimate.