| Abstract: |
| 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,
$$
R_i(T):=H\!\left(P^{1:i}_{[0,T]}\,\middle|\,P^{1:i-1}_{[0,T]}\otimes \bar P_{[0,T]}\right)
\ \lesssim\ \frac{1}{i-1}, \qquad i\ge2,
$$
where $P^{1:i}_{[0,T]}$ 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
$$
H\left(P^{1:N}_{[0,T]}\,\middle|\,\bar P_{[0,T]}^{\otimes N}\right)\ \lesssim\ \log N,
$$
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 $N^{-1/2}$ 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. |
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