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

TL;DR

This paper analyzes a sequential interacting diffusion system, proving entropy decay, convergence to McKean--Vlasov equations, and establishing Gaussian fluctuations with explicit feedback corrections.

Contribution

It introduces a novel sequential mean-field approximation framework with sharp entropy decay and Gaussian fluctuation results, including explicit feedback in the limiting SPDE.

Findings

Entropy decay rate of 1/i-1 for incremental path-space measures

Logarithmic bound on total relative entropy for N particles

Empirical measure converges at the N^{-1/2} scale with Gaussian fluctuations

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