Ruicheng Cheng | Searcharxiv

arXiv2026

Propagation of Chaos and Gaussian Fluctuations for Mean-field Coupled Maps

We study a class of discrete-time interacting particle systems arising from mean-field coupled maps. We establish a Dobrushin-type estimate, a relative entropy estimate and Central Limit Theorem (CLT) type results for this class of systems. First we control the growth of the 1-Wasserstein distance between the empirical measures and the limiting distributions. Then we use the relative entropy method to show the propagation of chaos. Finally we consider the asymptotic behavior of the fluctuations for the empirical measures and prove that the sequence of fluctuation processes converges in distribution to some Gaussian process, where we establish both qualitative and quantitative results.

math.PR↗

arXiv2026

On the kinetic equation arising from the large-scale limit of the Cucker-Smale model

We propose a large-scale scaling viewpoint for deriving mesoscopic dynamics from interacting particle systems and apply it to the Cucker--Smale flocking model. In contrast with the classical mean-field regime leading to the Vlasov-type Cucker--Smale equation with spatially nonlocal (convolution) alignment force, our scaling yields a kinetic equation whose alignment field becomes local in space and nonlocal only in velocity. For the spatially homogeneous case, we obtain an explicit solution and derive quantitative flocking rates. For the spatially inhomogeneous equation we establish a local well-posedness in $W^{1,\infty}$ and in $C_b^{1,\alpha}$, highlighting the additional difficulties caused by the absence of a convolution structure. Moreover, for sufficiently small interaction strength we present a global well-posedness and a forward-in-time $L^1$ asymptotic completeness property. Finally, we investigate mono-kinetic solutions and exhibit finite-time blow-up scenarios.

math.AP↗