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Propagation of Chaos for Derivatives of McKean-Vlasov Stochastic Differential Equations and Applications

Xiao-Yu Zhao

TL;DR

This work advances the theory of propagation of chaos by establishing derivative-level convergence for McKean–Vlasov SDEs, addressing both directional and Malliavin derivatives of mean-field and finite-particle systems. It develops a rigorous framework for derivative flows, proves their convergence with explicit rates, and shows sharp rates in favorable regimes. A key contribution is the finite-particle approximation of the intrinsic derivative of the distribution via a Bismut-type formula, with quantitative error bounds. Collectively, the results enhance gradient-based analyses and numerical methods for McKean–Vlasov dynamics by quantifying how derivative information propagates from particle systems to the mean-field limit. The work thus provides both theoretical insights and practical tools for sensitivity analysis in mean-field stochastic dynamics.

Abstract

As an enhanced version of existing results on Kac's propagation of chaos, which describes the convergence of mean-field particle systems to the independent McKean-Vlasov particle system as the number of particles tends to infinity, we prove the convergence at the level of derivatives with respect to the perturbations of initial value and noise, together with explicit convergence rates that can be sharp. As a consequence, the intrinsic derivative for the distribution of a single particle converges to that of an independent McKean-Vlasov SDE.

Propagation of Chaos for Derivatives of McKean-Vlasov Stochastic Differential Equations and Applications

TL;DR

This work advances the theory of propagation of chaos by establishing derivative-level convergence for McKean–Vlasov SDEs, addressing both directional and Malliavin derivatives of mean-field and finite-particle systems. It develops a rigorous framework for derivative flows, proves their convergence with explicit rates, and shows sharp rates in favorable regimes. A key contribution is the finite-particle approximation of the intrinsic derivative of the distribution via a Bismut-type formula, with quantitative error bounds. Collectively, the results enhance gradient-based analyses and numerical methods for McKean–Vlasov dynamics by quantifying how derivative information propagates from particle systems to the mean-field limit. The work thus provides both theoretical insights and practical tools for sensitivity analysis in mean-field stochastic dynamics.

Abstract

As an enhanced version of existing results on Kac's propagation of chaos, which describes the convergence of mean-field particle systems to the independent McKean-Vlasov particle system as the number of particles tends to infinity, we prove the convergence at the level of derivatives with respect to the perturbations of initial value and noise, together with explicit convergence rates that can be sharp. As a consequence, the intrinsic derivative for the distribution of a single particle converges to that of an independent McKean-Vlasov SDE.
Paper Structure (12 sections, 11 theorems, 134 equations)

This paper contains 12 sections, 11 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Classical results
  4. Derivative Flows
  5. Directional Derivative Flows
  6. Malliavin derivative flows
  7. Propagation of chaos for derivative flows
  8. Propagation of chaos for Directional derivative
  9. Propagation of chaos for Malliavin derivative
  10. Application: Propagation of chaos for intrinsic derivative
  11. Proof of Lemma \ref{['poc']}
  12. Extensions of Lemmas \ref{['gf1']} and \ref{['P3.1']}

Key Result

Lemma 2.1

Assume (H) for $k\geq1$. Then for any initial values $X_0\in L^k(\Omega\rightarrow \mathbb R^d,\mathscr F_0,\mathbb P)$, E2 has a unique solution $(X_t^{i,N})_{1\leq i\leq N}\in L^k(\Omega\rightarrow \mathbb R^d,\mathscr F_0,\mathbb P)$, which satisfies that for any $m\geq1$, there exists a constan Consequently, Moreover, If further assume that $\mu=\mathscr L_{X_0}\in\mathscr P_{q}$ for some $

Theorems & Definitions (22)

  • Lemma 2.1
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • Theorem 3.1
  • ...and 12 more