Quantitative Convergence and Gaussian Fluctuations for Sequential Interacting Diffusions via Incremental Relative Entropy

We study a sequential system of interacting diffusions in which particle $i$ interacts only with its predecessors through the empirical measure ${\mu }_t^{i-1}$, yielding a directed, non-exchangeable mean-field approximation of a McKean–Vlasov diffusion. Under bounded coefficients and a non-degenerate constant diffusion, we prove the sharp decay of incremental path-space relative entropies, $$R_i(T):=H(P_{[0,T]}^{1:i}\mathrm{\mid }{P}_{[0,T]}^{1:i-1}\otimes {\bar{P}}_{[0,T]})\lesssim \frac{1}{i-1},i\ge 2,$$ where $P_{[0,T]}^{1:i}$ is the law of the first $i$ particle paths and ${\bar{P}}_{[0,T]}$ the McKean–Vlasov path law. Summing the increments yields the global estimate $$H\left(P_{[0,T]}^{1:N}\mathrm{\mid }{\bar{P}}_{[0,T]}^{\otimes N}\right)\lesssim \log N,$$ together with quantitative decoupling bounds for tail blocks. As a consequence, the empirical measure converges to the McKean–Vlasov equation in negative Sobolev topologies at the canonical $N^{-1\mathrm{/}2}$ scale. We also establish a Gaussian fluctuation limit for the fluctuation measure, where the sequential architecture produces an explicit feedback correction in the limiting linear SPDE. Our proofs rely on a Girsanov representation, a martingale-difference replacement of predecessor empirical measures by average conditional measures, and an upper-envelope argument.