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Stability of Kantorovich potentials via heat kernel regularization

  • Bang-Xian Han
  • Zhuo-Nan Zhu
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Abstract

This survey presents heat kernel regularization as a method for quantitative stability of Kantorovich potentials for the quadratic transport cost. We give a complete new heat kernel proof on Heisenberg groups equipped with the Carnot--Carathéodory distance. For source densities bounded above and away from zero on bounded John domains, we obtain L2 stability with rate W11/2 for arbitrary targets in a fixed compact set, including atomic measures. We also establish a new L2 stability estimate on finite-dimensional RCD spaces, with the optimal dimension-independent exponent 1/2.

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Cite

BibTeXRIS
BibTeX
@misc{han2026stability,
  title = {{Stability of Kantorovich potentials via heat kernel regularization}},
  author = {Bang-Xian Han and Zhuo-Nan Zhu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.39213},
  eprint = {2609.39213},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39213},
}
RIS
TY  - UNPB
TI  - Stability of Kantorovich potentials via heat kernel regularization
AU  - Bang-Xian Han
AU  - Zhuo-Nan Zhu
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.39213
UR  - https://arxiv.org/abs/2609.39213
AB  - This survey presents heat kernel regularization as a method for quantitative stability of Kantorovich potentials for the quadratic transport cost. We give a complete new heat kernel proof on Heisenberg groups equipped with the Carnot--Carathéodory distance. For source densities bounded above and away from zero on bounded John domains, we obtain L^2 stability with rate W_1^1/2 for arbitrary targets in a fixed compact set, including atomic measures. We also establish a new L^2 stability estimate on finite-dimensional RCD spaces, with the optimal dimension-independent exponent 1/2.
ER  -