Quantitative Propagation of Chaos and Fluctuations for Kinetic McKean--Vlasov SDEs with Singular Interaction Kernels

We prove a quantitative propagation of chaos estimate and a central limit theorem for the particle system associated with a class of degenerate kinetic McKean--Vlasov SDEs with external drifts and singular interaction kernels in Kato's class.

In particular, the interaction kernel can be in the mixed $L^q_tL^{p_v}_vL^{p_x}_x$-space, where $\frac2q+\frac{3d}{p_x}+\frac d{p_v}<1$.

For the associated $N$-particle system, we obtain a path-space relative entropy bound of order $k/N$ for the first $k$ particles, assuming only entropic chaoticity of the initial data.

The key ingredients are kinetic Krylov--Khasminskii estimates and a conditional Hilbert-space subgaussian estimate for empirical interaction fields.

For the CLT, we also prove a Berry--Esseen-type bound for finite-dimensional projections.