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Quantitative Diffusive Limits for Singular Nonlocal Transport

arXiv · AI, language, vision and robotics · article · Sep 10, 2026 · UTC

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 als

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