Quantitative Diffusive Limits for Singular Nonlocal Transport

Abstract: We study the nonlocal continuity equation [ \partial_tμb =\operatorname{div}!\left( μ_b\nabla\log\bigl((I-b^2Δ)^{-1}μ_b\bigr) \right) ] on a closed connected Riemannian manifold. For smooth strictly positive initial data, we prove that as $b \to 0$, its global solution converges to heat flow $μ(t)$ at the sharp, uniform-in-time rate [ \sup{t\ge0}|μb(t)-μ(t)|{L^1}\le Cb^2. ] The key estimate is the uniform dissipation of a $b$-weighted higher-order resolvent energy, which yields exponential relaxation despite the absence of a Wasserstein gradient-flow structure. On the circle, we also analyze the corresponding deterministic $N$-particle dynamics. A weak--strong modulated energy argument gives [ \mathbb E!\left[ \sup_{t\ge0}W_1(μ_b^N(t),μ_b(t)) \right] \le C(Nb)^{-1/2} ] for iid initialization. Consequently, the choice $b\asymp N<sup>{-1/5}$ approximates heat flow uniformly in time at rate $N<sup>{-2/5}$.