Quantitative Diffusive Limits for Singular Nonlocal Transport
Abstract
We study the nonlocal continuity equation ∂tμb =div ( μb∇log((I−b2Δ)−1μb) ) on a closed connected Riemannian manifold. For smooth strictly positive initial data, we prove that as b → 0, its global solution converges to heat flow μ(t) at the sharp, uniform-in-time rate supt≥0|μb(t)−μ(t)|L1≤ Cb2. 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 𝔼 [ supt≥0W1(μbN(t),μb(t)) ] ≤ C(Nb)−1/2 for iid initialization. Consequently, the choice basymp N−1/5 approximates heat flow uniformly in time at rate N−2/5.
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BibTeX
@misc{agazzi2026quantitative,
title = {{Quantitative Diffusive Limits for Singular Nonlocal Transport}},
author = {Andrea Agazzi and Giuseppe Bruno and Federico Pasqualotto and Philippe Rigollet},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11837},
eprint = {2609.11837},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11837},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quantitative Diffusive Limits for Singular Nonlocal Transport
AU - Andrea Agazzi
AU - Giuseppe Bruno
AU - Federico Pasqualotto
AU - Philippe Rigollet
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11837
UR - https://arxiv.org/abs/2609.11837
AB - We study the nonlocal continuity equation ∂_tμ_b =div ( μ_b∇log((I−b^2Δ)^−1μ_b) ) on a closed connected Riemannian manifold. For smooth strictly positive initial data, we prove that as b → 0, its global solution converges to heat flow μ(t) at the sharp, uniform-in-time rate sup_t≥0|μ_b(t)−μ(t)|_L^1≤ 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 𝔼 [ sup_t≥0W_1(μ_b^N(t),μ_b(t)) ] ≤ C(Nb)^−1/2 for iid initialization. Consequently, the choice basymp N^−1/5 approximates heat flow uniformly in time at rate N^−2/5.
ER -