Quantitative Diffusive Limits for Singular Nonlocal Transport

This paper establishes quantitative diffusive limits for a class of singular nonlocal continuity equations on closed Riemannian manifolds. The authors prove that as the smoothing parameter $b \to 0$, solutions converge to standard heat flow at a sharp, uniform-in-time $L^1$ rate of $Cb^2$, leveraging uniform dissipation of a higher-order resolvent energy to circumvent the lack of a Wasserstein gradient-flow structure. For deterministic $N$-particle dynamics on the circle, setting $b \asymp N^{-1/5}$ yields an approximation rate of $N^{-2/5}$, providing a rigorous microscopic-to-continuum error estimate.