Stability of optimal transport on sub-Riemannian manifolds
Abstract
We study the stability of optimal transport for the squared Carnot--Carathéodory distance. We prove the sharp W11/4 stability estimate for optimal transport maps on complete Riemannian and fat sub-Riemannian manifolds, assuming bounded source support and an upper density bound. On John domains with two-sided density bounds, we obtain logarithmic L2 stability of Kantorovich potentials on complete smooth sub-Riemannian manifolds that are equiregular near the source, with arbitrary compact targets. For potentials, we obtain every Hölder exponent 0<α<1/2 on two-generating structures, and the exponent 1/2 in the Riemannian and fat settings and on all step-two Carnot groups. Three-atom examples show that the map and potential exponents with respect to W1 cannot exceed 1/4 and 1/2, respectively, on any smooth sub-Riemannian manifold of horizontal rank at least two. We also construct smooth transport families whose target variation of order t in W1 produces map variation of order t1/s in an equiregular region of step s. The map estimates follow from coercivity of the duality defect, while the potential estimates follow from quantitative concavity of the Kantorovich functional.
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BibTeX
@misc{han2026stability,
title = {{Stability of optimal transport on sub-Riemannian manifolds}},
author = {Bang-Xian Han and Zhuo-Nan Zhu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09688},
eprint = {2610.09688},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09688},
}
RIS
TY - UNPB
TI - Stability of optimal transport on sub-Riemannian manifolds
AU - Bang-Xian Han
AU - Zhuo-Nan Zhu
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09688
UR - https://arxiv.org/abs/2610.09688
AB - We study the stability of optimal transport for the squared Carnot--Carathéodory distance. We prove the sharp W_1^1/4 stability estimate for optimal transport maps on complete Riemannian and fat sub-Riemannian manifolds, assuming bounded source support and an upper density bound. On John domains with two-sided density bounds, we obtain logarithmic L^2 stability of Kantorovich potentials on complete smooth sub-Riemannian manifolds that are equiregular near the source, with arbitrary compact targets. For potentials, we obtain every Hölder exponent 0<α<1/2 on two-generating structures, and the exponent 1/2 in the Riemannian and fat settings and on all step-two Carnot groups. Three-atom examples show that the map and potential exponents with respect to W_1 cannot exceed 1/4 and 1/2, respectively, on any smooth sub-Riemannian manifold of horizontal rank at least two. We also construct smooth transport families whose target variation of order t in W_1 produces map variation of order t^1/s in an equiregular region of step s. The map estimates follow from coercivity of the duality defect, while the potential estimates follow from quantitative concavity of the Kantorovich functional.
ER -