[Paper Review] Propagation of chaos for Hölder continuous interaction kernels via Glivenko-Cantelli

Thomas Holding|arXiv (Cornell University)|Aug 9, 2016

Stochastic processes and financial applications20 references16 citations

TL;DR

This paper introduces a novel technique combining empirical process theory and stochastic flows to establish quantitative propagation of chaos for interacting particle systems with Hölder continuous interaction kernels, even when the kernel is nowhere Lipschitz. The method achieves polynomial convergence rates in $N$ without requiring specially prepared initial data, relying crucially on the presence of noise, and extends to second-order Langevin systems with Hölder exponent $>2/3$. The key contribution is a Glivenko-Cantelli-type law of large numbers for SDEs over a class of Hölder continuous vector fields.

ABSTRACT

We develop a new technique for establishing quantitative propagation of chaos for systems of interacting particles. Using this technique we prove propagation of chaos for diffusing particles whose interaction kernel is merely Hölder continuous, even at long ranges. Moreover, we do not require specially prepared initial data. On the way, we establish a law of large numbers for SDEs that holds over a class of vector fields simultaneously. The proofs bring together ideas from empirical process theory and stochastic flows.

Motivation & Objective

• To establish quantitative propagation of chaos for diffusing particles with Hölder continuous interaction kernels, even when the kernel is nowhere Lipschitz.
• To remove the need for specially prepared initial data, which is required in prior coupling-based methods.
• To develop a Glivenko-Cantelli-type law of large numbers for SDEs over a class of bounded Hölder continuous vector fields.
• To extend the analysis to second-order Langevin systems with Hölder continuous kernels and a minimal Hölder exponent of $2/3$.
• To unify empirical process theory and stochastic flows to overcome the limitations of classical coupling methods in non-Lipschitz settings.

Proposed method

• The method employs a new framework that combines empirical process theory with the analysis of stochastic flows to control the empirical measure of particle systems.
• It establishes a Glivenko-Cantelli theorem for SDEs driven by vector fields in a class of bounded Hölder continuous functions, ensuring uniform convergence over the class.
• The approach uses anisotropic Besov spaces and entropy estimates to control the complexity of the vector field class, enabling uniform bounds on the empirical measure.
• For the first-order system, the method proves that the empirical measure $\mu_t^N$ converges to the solution $f_t$ of the mean-field Fokker-Planck equation at a polynomial rate in $N$.
• For the second-order system, the method applies under the condition that the interaction kernel is Hölder continuous with exponent $>2/3$, due to the degeneracy of the generator.
• The proof relies on a novel entropy bound on the class of vector fields, leveraging the parabolic anisotropy and Sobolev embedding to control the approximation error.

Experimental results

Research questions

• RQ1 Can propagation of chaos be established for interacting particle systems with Hölder continuous interaction kernels that are not Lipschitz anywhere?
• RQ2 Can the convergence rate be quantified without requiring specially prepared initial data?
• RQ3 Is it possible to derive a Glivenko-Cantelli-type law of large numbers for SDEs driven by a class of Hölder continuous vector fields?
• RQ4 How does the degeneracy of the generator in second-order systems affect the required regularity of the interaction kernel?
• RQ5 Can the coupling method's limitations in non-Lipschitz settings be overcome using empirical process theory and stochastic flows?

Key findings

• The paper establishes quantitative propagation of chaos for the first-order system with Hölder continuous interaction kernels, achieving convergence rate $O(N^{-\gamma})$ for some $\gamma > 0$, independent of initial data preparation.
• The method does not require the interaction kernel to be Lipschitz, even at long ranges, and applies to kernels that are nowhere differentiable.
• A new Glivenko-Cantelli theorem is proven for SDEs over a class of bounded Hölder continuous vector fields, with uniform convergence in $N$ and over the class of vector fields.
• For second-order Langevin systems, propagation of chaos holds under the condition that the kernel is Hölder continuous with exponent greater than $2/3$, due to hypoelliptic regularity constraints.
• The convergence rate in the second-order case is also polynomial in $N$, with the exponent depending on the Hölder exponent and dimension.
• The analysis relies on entropy bounds in anisotropic Besov spaces, with the key estimate $H(\varepsilon, \mathcal{C}, \|\cdot\|_{L^{-r,\infty}}) \leq C\varepsilon^{-d/s}$, which controls the complexity of the vector field class.