Quantitative propagation of chaos for particle systems with bounded kernels and multiplicative noise

Abstract: We prove the quantitative propagation of chaos for stochastic particle systems with interaction in both the drift and the diffusion coefficients, provided the drift kernel is bounded and free of Lipschitz or smoothness assumptions. Our proof is based on the relative entropy framework of Jabin and Wang \cite{JW2018}, and applies and extends their work on the exponential laws of large numbers. We extend one of their exponential laws of large numbers from the drift to the diffusion kernel to handle the error term arising from multiplicative noise in the entropy evolution equation. Proving this extension relies on a dynamic combinatorial analysis.

• The paper introduces a quantitative propagation of chaos bound in relative entropy for particle systems with irregular drift and interaction-dependent multiplicative noise.
• It employs a refined relative entropy method to decouple errors from drift and variable diffusion, achieving a uniform 1/N convergence rate.
• The work advances the analysis of mean-field models, impacting applications like stochastic synchronization, flocking, and particle algorithms in machine learning.

Quantitative Propagation of Chaos for Particle Systems with Bounded Kernels and Multiplicative Noise

Introduction and Context

This paper (2604.11084) addresses the quantitative propagation of chaos for mean-field interacting particle systems with bounded interaction kernels and non-constant, interaction-dependent (multiplicative) noise. The considered systems consist of $N$ indistinguishable particles on the torus $\mathbb{T}^d$ described by SDEs with both drift and diffusion coefficients depending on the empirical distribution. The main novelty is the extension of the quantitative relative entropy method—previously established for constant or regular diffusion coefficients—to the case of bounded, possibly highly irregular drift kernels and non-constant, interaction-dependent diffusion kernels.

Prior work on propagation of chaos for particle systems typically required either globally Lipschitz or smooth coefficients, or was limited to constant (additive) noise. This restriction excluded important models with multiplicative or signal-dependent noise arising in, e.g., stochastic synchronization (Kuramoto) and flocking (Cucker–Smale–type) models, stochastic control, mean-field games, and the analysis of interacting particle algorithms in ML.

Model and Main Assumptions

The particle system under consideration evolves according to

$$d X^i_t = \frac{1}{N} \sum_{k=1}^N K(X^i_t - X^k_t)\, dt + \frac{\sqrt{2}}{N} \sum_{k=1}^N \sigma(X^i_t - X^k_t)\, dB^i_t, \qquad i=1,\ldots,N,$$

where the drift kernel $K:\mathbb{T}^d\to\mathbb{R}^d$ is bounded, i.e., $K \in L^\infty$, with weak regularity $div\,K \in \dot{W}^{-1,\infty}$, and the diagonal diffusion kernel $\sigma$ is $W^{2,\infty}$ and uniformly elliptic. Both coefficients can lack global Lipschitz or smoothness properties, which marks a departure from existing literature.

The mean-field (McKean–Vlasov) limit is given by a nonlinear, nonlocal PDE: $$\partial_t \bar\rho + \nabla \cdot \left((K*\bar\rho) \bar\rho\right) = \sum_{\alpha=1}^d \partial_{x^\alpha}^2 \left(\bar\rho\,[\sigma_{\alpha\alpha}*\bar\rho(x)]^2\right),$$ where $\bar\rho$ is the one-particle density. This introduces a nontrivial, multiplicative, and distribution-dependent noise term at the PDE level.

Main Theorem and Quantitative Estimates

The central result is a non-asymptotic, quantitative propagation of chaos bound in relative entropy. Precisely, letting $\mathbb{T}^d$0 denote the joint law of the $\mathbb{T}^d$1-particle system and $\mathbb{T}^d$2 the product law from the mean-field PDE, the (rescaled) relative entropy $\mathbb{T}^d$3 satisfies: $\mathbb{T}^d$4 where $\mathbb{T}^d$5 is universal, and $\mathbb{T}^d$6 depends quantitatively on the norms of $\mathbb{T}^d$7, $\mathbb{T}^d$8, $\mathbb{T}^d$9, and on the regularity of the limiting density $$d X^i_t = \frac{1}{N} \sum_{k=1}^N K(X^i_t - X^k_t)\, dt + \frac{\sqrt{2}}{N} \sum_{k=1}^N \sigma(X^i_t - X^k_t)\, dB^i_t, \qquad i=1,\ldots,N,$$0. The explicit dependence includes contributions from both drift and diffusion, and crucially, the bound is uniform in $$d X^i_t = \frac{1}{N} \sum_{k=1}^N K(X^i_t - X^k_t)\, dt + \frac{\sqrt{2}}{N} \sum_{k=1}^N \sigma(X^i_t - X^k_t)\, dB^i_t, \qquad i=1,\ldots,N,$$1 up to the rate $$d X^i_t = \frac{1}{N} \sum_{k=1}^N K(X^i_t - X^k_t)\, dt + \frac{\sqrt{2}}{N} \sum_{k=1}^N \sigma(X^i_t - X^k_t)\, dB^i_t, \qquad i=1,\ldots,N,$$2. Through the Csiszár–Kullback–Pinsker inequality and entropy subadditivity, this yields strong quantitative convergence for finite-dimensional marginals.

This result is achieved under weak regularity assumptions on $$d X^i_t = \frac{1}{N} \sum_{k=1}^N K(X^i_t - X^k_t)\, dt + \frac{\sqrt{2}}{N} \sum_{k=1}^N \sigma(X^i_t - X^k_t)\, dB^i_t, \qquad i=1,\ldots,N,$$3 (bounded, with divergence in negative Sobolev space) and only uniform ellipticity and $$d X^i_t = \frac{1}{N} \sum_{k=1}^N K(X^i_t - X^k_t)\, dt + \frac{\sqrt{2}}{N} \sum_{k=1}^N \sigma(X^i_t - X^k_t)\, dB^i_t, \qquad i=1,\ldots,N,$$4 regularity on $$d X^i_t = \frac{1}{N} \sum_{k=1}^N K(X^i_t - X^k_t)\, dt + \frac{\sqrt{2}}{N} \sum_{k=1}^N \sigma(X^i_t - X^k_t)\, dB^i_t, \qquad i=1,\ldots,N,$$5, which are strictly weaker than classical (global Lipschitz or smoothness) conditions.

Methodology and Technical Contributions

The analysis employs a relative entropy method, building on the framework developed by Jabin & Wang. The entropy dissipation equation is derived for the $$d X^i_t = \frac{1}{N} \sum_{k=1}^N K(X^i_t - X^k_t)\, dt + \frac{\sqrt{2}}{N} \sum_{k=1}^N \sigma(X^i_t - X^k_t)\, dB^i_t, \qquad i=1,\ldots,N,$$6-particle system, identifying principal errors arising from the non-constant, multiplicative diffusion term. A crucial challenge is that the corresponding error term in the entropy evolution—absent in constant diffusion—requires a new treatment.

The main technical advance is a new exponential law of large numbers (large deviation estimate) for the interaction-dependent diffusion term, generalizing and extending prior exponential concentration results originally tailored for the drift. This involves an intricate combinatorial analysis to handle nonlinear functionals involving triple interactions, and exploits cancellation properties in the symmetric setting.

The core steps are:

• Derivation of the evolution inequality for $$d X^i_t = \frac{1}{N} \sum_{k=1}^N K(X^i_t - X^k_t)\, dt + \frac{\sqrt{2}}{N} \sum_{k=1}^N \sigma(X^i_t - X^k_t)\, dB^i_t, \qquad i=1,\ldots,N,$$7, separating classical drift-related errors from new nontrivial contributions due to variable diffusion.
• Application of combinatorics and cancellation rules to control higher-order terms, circumventing the direct Lipschitz/smoothness requirements on $$d X^i_t = \frac{1}{N} \sum_{k=1}^N K(X^i_t - X^k_t)\, dt + \frac{\sqrt{2}}{N} \sum_{k=1}^N \sigma(X^i_t - X^k_t)\, dB^i_t, \qquad i=1,\ldots,N,$$8.
• Use of a quantitative entropy–information inequality to relate control of three-particle errors to overall entropy dissipation.
• Deployment of a Grönwall argument to close the propagation of chaos estimate, yielding explicit exponential-in-time control with the desired $$d X^i_t = \frac{1}{N} \sum_{k=1}^N K(X^i_t - X^k_t)\, dt + \frac{\sqrt{2}}{N} \sum_{k=1}^N \sigma(X^i_t - X^k_t)\, dB^i_t, \qquad i=1,\ldots,N,$$9 scaling.

The existence and uniqueness of (classical) solutions to both the limit PDE and the particle-level master equation are proven for sufficiently regular initial data, using an iterative argument and a priori Sobolev bounds.

Numerical and Analytical Strengths

The results are robust: the derived estimate is strong in the sense of relative entropy, yields an explicit finite-$K:\mathbb{T}^d\to\mathbb{R}^d$0 convergence rate, and does not require the drift or diffusion to be globally Lipschitz or smooth. This broadens the class of mean-field models accessible to quantitative analysis, including those with oscillatory or highly irregular interaction kernels, provided only boundedness and weak divergence control. The results are also stable under oscillatory or degenerate perturbations in the drift and allow for significant modeling generality in applications involving multiplicative noise.

Implications and Directions

This work closes a gap in the propagation of chaos literature at the interface of bounded singularity analysis and variable-coefficient diffusion, opening access to a larger class of interacting stochastic systems relevant in statistical physics, stochastic modeling, and applied computational schemes (including interacting particle algorithms and stochastic mean-field game theory).

On the theoretical side, the paper demonstrates that relative entropy-based methods are flexible enough to handle variable, even nonlinear, diffusion structures provided their symmetry and combinatorial cancellation are exploited. This may encourage further work to lower the regularity requirements on $K:\mathbb{T}^d\to\mathbb{R}^d$1, consider interaction-driven common noise, or treat more singular kernels (e.g., Coulomb) by refining the combinatorial argument, possibly via concentration-of-measure or sharp large deviation inequalities.

In practice, the developed entropy dissipation and combinatorial methods can be adapted or extended to analyze high-dimensional stochastic interacting systems, complex agent-based models, and signal-dependent sampling schemes in machine learning, where multiplicative noise plays a crucial role.

Conclusion

The paper provides the first quantitative propagation of chaos result for mean-field particle systems with bounded, possibly highly irregular drift and interaction-dependent, multiplicative noise, establishing explicit non-asymptotic entropy convergence rates under minimal regularity assumptions. The methodological advancement—an extension of the relative entropy method via a novel exponential law of large numbers for variable diffusion—sets a new benchmark for the quantitative analysis of nonlinear and distribution-dependent interacting stochastic systems (2604.11084).