Quantitative Diffusive Limits for Singular Nonlocal Transport

• URL: http://arxiv.org/abs/2609.11837v1
• Date: Thu, 10 Sep 2026 17:29:01 GMT
• Title: Quantitative Diffusive Limits for Singular Nonlocal Transport
• Abstract summary: We study the nonlocal continuity equation [ partial_t_b =operatornamediv!!left( _bnablalogbigl((I-b2)-1_bbigr).<n>For smooth strictly positive initial data, we prove that as $b to 0$, its global solution converges to heat flow $(t)$ at the sharp, uniform-in-time rate.
• Score: 11.229917469094966
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study the nonlocal continuity equation \[ \partial_tμ_b =\operatorname{div}\!\left( μ_b\nabla\log\bigl((I-b^2Δ)^{-1}μ_b\bigr) \right) \] 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 \[ \sup_{t\ge0}\|μ_b(t)-μ(t)\|_{L^1}\le Cb^2. \] 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 $b\asymp N^{-1/5}$ approximates heat flow uniformly in time at rate $N^{-2/5}$.

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