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Entropy-cost type Propagation of Chaos for Mean Field Particle System with Bounded Measurable Interaction

Xing Huang

TL;DR

This work establishes quantitative entropy-cost type propagation of chaos for mean-field particle systems with bounded measurable interaction, under initial data converging in the $L^1$-transportation metric $\\mathbb W_1^\\Psi$ to the McKean–Vlasov law. It extends the analysis to systems with common noise, proving conditional propagation of chaos with the same entropy-cost structure in terms of $\\mathbb W_1^\\Psi_\\eta$, capturing the impact of environment noise. The results rely on Wang's Harnack inequalities, Girsanov transforms, and two key lemmas connecting $N$-particle costs to single-particle initial data, enabling explicit rates in $N$ and $t$ with singular but bounded interactions. Additionally, the paper proves strong well-posedness for conditional McKean–Vlasov SDEs via a fixed-point approach on a space of measure-valued processes, ensuring the theoretical soundness of the conditional propagation results and expanding the toolbox for analyzing distribution-dependent dynamics under noisy environments.

Abstract

In this paper, the quantitative entropy-cost type propagation of chaos for mean field interacting particle system is obtained, where the interaction is only assumed to be bounded measurable and the initial distribution of a single particle converges to that of the corresponding McKean-Vlasov SDEs in $L^1$-transportation cost $\W_1^Ψ$ induced by some cost function $Ψ$. The mean field interacting particle system with common noise is also investigated and the quantitative entropy-cost type conditional propagation of chaos is established. The results weaken the initial assumption of the existing entropy-entropy type propagation of chaos in some sense.

Entropy-cost type Propagation of Chaos for Mean Field Particle System with Bounded Measurable Interaction

TL;DR

This work establishes quantitative entropy-cost type propagation of chaos for mean-field particle systems with bounded measurable interaction, under initial data converging in the -transportation metric to the McKean–Vlasov law. It extends the analysis to systems with common noise, proving conditional propagation of chaos with the same entropy-cost structure in terms of , capturing the impact of environment noise. The results rely on Wang's Harnack inequalities, Girsanov transforms, and two key lemmas connecting -particle costs to single-particle initial data, enabling explicit rates in and with singular but bounded interactions. Additionally, the paper proves strong well-posedness for conditional McKean–Vlasov SDEs via a fixed-point approach on a space of measure-valued processes, ensuring the theoretical soundness of the conditional propagation results and expanding the toolbox for analyzing distribution-dependent dynamics under noisy environments.

Abstract

In this paper, the quantitative entropy-cost type propagation of chaos for mean field interacting particle system is obtained, where the interaction is only assumed to be bounded measurable and the initial distribution of a single particle converges to that of the corresponding McKean-Vlasov SDEs in -transportation cost induced by some cost function . The mean field interacting particle system with common noise is also investigated and the quantitative entropy-cost type conditional propagation of chaos is established. The results weaken the initial assumption of the existing entropy-entropy type propagation of chaos in some sense.
Paper Structure (5 sections, 5 theorems, 112 equations)

This paper contains 5 sections, 5 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Mean field interacting particle system
  3. Extension to mean field interacting particle system with common noise
  4. Entropy-cost type conditional propagation of chaos
  5. Well-posedness for conditional McKean-Vlasov SDEs

Key Result

Theorem 2.1

Assume (A1)-(A2), $(X_0^{i,N})_{1\leq i\leq N}$ are i.i.d. and $\mathscr L_{X_0^{1,N}},\mathscr L_{X_0^{1}}\in \mathscr P_2(\mathbb R^d)$. Then there exists a constant $C>0$ depending on $d,\delta,T,\|\tilde{b}\|_\infty$ such that for any $\eta\in(0,1)$, $t\in(0,T]$ and $1\leq k\leq N$, where $\Psi_\eta(x,y)=|x-y|^{2\eta}+|x-y|^2, x,y\in\mathbb R^d$.

Theorems & Definitions (12)

  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • proof : Proof of Theorem \ref{['POC']}
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • ...and 2 more