Quantitative Convergence for Sequential Interacting Diffusions via Incremental Relative Entropy

We study a lower-triangular system of interacting diffusions in which particle \(i\) interacts only with its predecessors through the empirical measure \(\mu^{i-1}_t\).

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, $$ R_i(T) = H \left(P^{1:i}_{[0,T]}\,\middle|\,P^{1:i-1}_{[0,T]}\otimes \bar P_{[0,T]}\right), $$ 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.