Abstract: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.
From: Zhenfu Wang [view email]
[v1]
Mon, 2 Feb 2026 04:57:39 UTC (60 KB)
[v2]
Thu, 23 Jul 2026 10:25:44 UTC (46 KB)
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