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Sharp local propagation of chaos for mean field particles with $W^{-1,\infty}$ kernels

Songbo Wang

TL;DR

This work addresses quantitative propagation of chaos for $N$ diffusive particles with singular mean-field interactions in the $W^{-1,fty}$ class. It develops two complementary hierarchical frameworks: an entropic hierarchy using relative entropy and Fisher information, and an $L^2$/Dirichlet-energy hierarchy, to compare $k$-particle marginals with the tensorized mean-field law. The entropic approach yields global-in-time sharp bounds $H^k_t \,=\, O\left( k^2/N^2 \right)$ (and exponential decay in the 2D viscous vortex regime under weak coupling), while the $L^2$ method provides short-time $D^k_t \,=\, O\left( k^2/N^2 \right)$ controls for fixed $k$ without smallness assumptions. The results extend Lacker to $W^{-1,\infty}$ interactions and provide a robust local PoC theory for singular kernels, including a detailed treatment of inner/outer interaction terms and a set of technical tools (transport inequalities, refined JW lemma, maximum principle). These contributions deepen understanding of mean-field limits in the presence of singular forces and offer new avenues for uniform-in-time results under structural conditions in 2D vortex-type models.

Abstract

We present two methods to obtain $O(1/N^2)$ local propagation of chaos bounds for $N$ diffusive particles in $W^{-1,\infty}$ mean field interaction. This extends the recent finding of Lacker [Probab. Math. Phys., 4(2):377-432, 2023] to the case of singular interactions. The first method is based on a hierarchy of relative entropies and Fisher informations, and applies to the 2D viscous vortex model in the high temperature regime. Time-uniform local chaos bounds are also shown in this case. In the second method, we work on a hierarchy of $L^2$ distances and Dirichlet energies, and derive the desired sharp estimates for the same model in short time without restrictions on the temperature.

Sharp local propagation of chaos for mean field particles with $W^{-1,\infty}$ kernels

TL;DR

This work addresses quantitative propagation of chaos for diffusive particles with singular mean-field interactions in the class. It develops two complementary hierarchical frameworks: an entropic hierarchy using relative entropy and Fisher information, and an /Dirichlet-energy hierarchy, to compare -particle marginals with the tensorized mean-field law. The entropic approach yields global-in-time sharp bounds (and exponential decay in the 2D viscous vortex regime under weak coupling), while the method provides short-time controls for fixed without smallness assumptions. The results extend Lacker to interactions and provide a robust local PoC theory for singular kernels, including a detailed treatment of inner/outer interaction terms and a set of technical tools (transport inequalities, refined JW lemma, maximum principle). These contributions deepen understanding of mean-field limits in the presence of singular forces and offer new avenues for uniform-in-time results under structural conditions in 2D vortex-type models.

Abstract

We present two methods to obtain local propagation of chaos bounds for diffusive particles in mean field interaction. This extends the recent finding of Lacker [Probab. Math. Phys., 4(2):377-432, 2023] to the case of singular interactions. The first method is based on a hierarchy of relative entropies and Fisher informations, and applies to the 2D viscous vortex model in the high temperature regime. Time-uniform local chaos bounds are also shown in this case. In the second method, we work on a hierarchy of distances and Dirichlet energies, and derive the desired sharp estimates for the same model in short time without restrictions on the temperature.
Paper Structure (18 sections, 11 theorems, 173 equations)

This paper contains 18 sections, 11 theorems, 173 equations.

Table of Contents

  1. Introduction and main results
  2. Proof of Theorems \ref{['thm:entropy']} and \ref{['thm:l2']}
  3. Setup and proof outline
  4. Two lemmas on inner interaction terms
  5. Control of the inner interaction terms
  6. Control of the regular part $A_2$
  7. Control of the singular part $A_1$
  8. Control of the outer interaction terms
  9. Control of the regular part $B_2$
  10. Control of the singular part $B_1$
  11. Conclusion of the proof
  12. ODE hierarchies
  13. Entropic hierarchy
  14. $L^2$ hierarchy
  15. Other technical results
  16. ...and 3 more sections

Key Result

Theorem 1

Let the main assumption hold. Suppose that the marginal relative entropies at the initial time satisfy for all $k \in [N]$, for some $C_0 \geqslant 0$. If $\lVert V\rVert_{L^{\infty}} < 1$, then for all $T > 0$, there exists $M$, depending on such that for all $t \in [0,T]$, If additionally $K_2 = 0$ and for all $t \geqslant 0$, for some $M_m \geqslant 0$ and $\eta > 0$, then for all $r$ such

Theorems & Definitions (21)

  • Theorem 1: Entropic PoC
  • Remark 1
  • Theorem 2: $L^2$ PoC
  • Remark 2
  • Remark 3
  • Lemma 3
  • Lemma 4
  • proof : Proof of Lemma \ref{['lem:internal-interaction-bounded']}
  • proof : Proof of Lemma \ref{['lem:internal-interaction-bounded-ibp']}
  • Proposition 5
  • ...and 11 more