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Propagation of chaos for moderately interacting particle systems related to singular kinetic McKean-Vlasov SDEs

Zimo Hao, Jean-Francois Jabir, Stéphane Menozzi, Michael Röckner, Xicheng Zhang

TL;DR

This work develops a quantitative propagation of chaos theory for moderately interacting particle systems that approximate singular kinetic McKean–Vlasov SDEs driven by isotropic ${\alpha}$-stable noise. By mollifying distributional drifts and employing an anisotropic Besov space framework aligned with kinetic scaling, it derives explicit weak and pathwise convergence rates that balance mollification and stable fluctuations. The main contributions include a Duhamel-based analysis for the mollified empirical measure, rigorous bounds on deterministic and martingale error terms, and the extension of these results to Coulomb and Riesz-type kernels, with explicit optimal rates in kinetic regimes. These results provide a rigorous foundation for particle methods in singular kinetic models and offer insights into the interplay between regularity, stability, and multiscale interactions in second-order stochastic systems.

Abstract

This work addresses the propagation of chaos properties in a class of moderately interacting particle systems for the approximation of singular kinetic McKean-Vlasov SDEs driven by alpha-stable processes.

Propagation of chaos for moderately interacting particle systems related to singular kinetic McKean-Vlasov SDEs

TL;DR

This work develops a quantitative propagation of chaos theory for moderately interacting particle systems that approximate singular kinetic McKean–Vlasov SDEs driven by isotropic -stable noise. By mollifying distributional drifts and employing an anisotropic Besov space framework aligned with kinetic scaling, it derives explicit weak and pathwise convergence rates that balance mollification and stable fluctuations. The main contributions include a Duhamel-based analysis for the mollified empirical measure, rigorous bounds on deterministic and martingale error terms, and the extension of these results to Coulomb and Riesz-type kernels, with explicit optimal rates in kinetic regimes. These results provide a rigorous foundation for particle methods in singular kinetic models and offer insights into the interplay between regularity, stability, and multiscale interactions in second-order stochastic systems.

Abstract

This work addresses the propagation of chaos properties in a class of moderately interacting particle systems for the approximation of singular kinetic McKean-Vlasov SDEs driven by alpha-stable processes.
Paper Structure (21 sections, 31 theorems, 389 equations)

This paper contains 21 sections, 31 theorems, 389 equations.

Table of Contents

  1. Introduction
  2. McKean-Vlasov SDEs
  3. Moderately interacting particle systems
  4. Anisotropic Besov space and kinetic semi-group
  5. Main results
  6. Convergence of empirical measure via Duhamel's formula
  7. Technical preliminaries on the anisotropic Besov spaces
  8. Proof of Theorem \ref{['main01']}
  9. Proof of Theorem \ref{['S1:main01']}
  10. Proof of the main technical Lemmas
  11. Proof of Lemma \ref{['lem:HN']}
  12. Proof of Lemma \ref{['lem:MN']}
  13. Applications to Coulomb and Riesz type interaction kernel (and a remark on the non-degenerate case)
  14. Weak wellposedness of \ref{['VPFP-McKV']}.
  15. Weak and moderate propagation of chaos for \ref{['VPFP-Particle']}.
  16. ...and 6 more sections

Key Result

Theorem 1.2

Suppose that (H) holds, and that either ${\boldsymbol{p}}_0>\boldsymbol{\alpha}$, or $(\alpha,{\boldsymbol{p}}_0)=(2,\boldsymbol{1})$. Let $\{\xi_i\}_{i=1}^N$ be, as in S1:00, a family of i.i.d. ${\mathbb R}^{2d}$-valued random variables with common law $\mu_0$, and assume that $\mu_0$ satisfies: Then, for any $\beta,\zeta>0$ with the following upper bounds and, for any $\varepsilon>0$ there i

Theorems & Definitions (64)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3: About the convergence rates.
  • Remark 1.4: About the sampling of the initial condition
  • Remark 1.5: About the integrability parameters ${\boldsymbol{p}}_b$ and ${\boldsymbol{p}}_0$
  • Theorem 2.1
  • Lemma 2.2: Bernstein's inequality
  • Corollary 2.3
  • proof
  • Proposition 2.4
  • ...and 54 more