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

Zhenfu Wang, Xianliang Zhao

Abstract

We study a sequential system of interacting diffusions in which particle $i$ interacts only with its predecessors through the empirical measure $μ_t^{i-1}$, 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.

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

Abstract

We study a sequential system of interacting diffusions in which particle interacts only with its predecessors through the empirical measure , 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, where is the law of the first particle paths and the McKean--Vlasov path law. Summing the increments yields the global estimate 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 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.
Paper Structure (24 sections, 32 theorems, 280 equations, 1 figure)

This paper contains 24 sections, 32 theorems, 280 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main results
  4. Related literature and proof strategy.
  5. Organization of the article
  6. Preliminaries
  7. Proof of the incremental relative entropy bound
  8. Quadratic identity for $R_i(t)$
  9. Replacement of the predecessor empirical measure
  10. Upper-envelope closure and tail chaos
  11. Global path-space entropy via Hardy's inequality
  12. Possible extension to systems with singular interactions.
  13. Convergence of empirical measure in negative Sobolev norms
  14. Kernel energy identity and a priori estimate
  15. Proof of Theorem \ref{['thm:empirical']}
  16. ...and 9 more sections

Key Result

Theorem 1

Assume that the diffusion matrix $\sigma$ is invertible and that the drift term $b$ and the interaction $K$ are bounded measurable. For $i\ge2$ set Then there exists $C_T>0$, depending only on $T$, $\|\sigma^{-1}\|$, $\|b\|_\infty$ and $\|K\|_\infty$, such that for all $t\in[0,T]$ and all $i\ge2$, In particular, if $M_0:=\sup_{i\ge2}M_{0,i}<\infty$, then $R_i(t)\le \frac{C_T}{i-1}(1+M_0)$ unifor

Figures (1)

  • Figure 1: Left: The classical fully-coupled system where every particle pair interacts, scaling as $O(N^2)$. Right: The sequential system where each particle interacts with a pre-computed summary statistic $\mu^{i-1}$ and updates it in $O(1)$ time, scaling as $O(N)$.

Theorems & Definitions (70)

  • Theorem 1
  • Remark 1.1
  • Corollary 1.1
  • Remark 1.2: Sharpness of Theorem \ref{['thm:incremental']}
  • Corollary 1.2
  • Remark 1.3
  • Theorem 2
  • Remark 1.4
  • Remark 1.5: Regularity assumption on the velocity field $v= K \star \bar{\rho}$
  • Theorem 3: Gaussian Fluctuations
  • ...and 60 more