Quantitative Convergence for Sequential Interacting Diffusions via Incremental Relative Entropy

Abstract

We study a lower-triangular system of interacting diffusions in which particle \(i\) interacts only with its predecessors through the empirical measure \(μ^{i-1}_t\). This gives a directed, non-exchangeable approximation of the same McKean--Vlasov diffusion as the classical exchangeable particle system. We introduce an incremental path-space relative entropy adapted to the causal structure, $$R_i(T)=H\left(P_{[0,T]}^{1:i}|P_{[0,T]}^{1:i-1}\otimes {\bar{P}}_{[0,T]}\right),$$ and prove the sharp estimate \(R_i(T)\lesssim (i-1)^{-1}\). Furthermore, we obtain convergence of the empirical measure to the McKean--Vlasov law at the canonical \(N^{-1/2}\) scale in negative Sobolev norms. The proof combines a Girsanov representation, a martingale-difference replacement of predecessor empirical measures by averaged conditional measures, an upper-envelope closure, and a negative Sobolev energy estimate.

35 pages, 2 figures. We split the first version to two parts. This is the first part