Year: 2026
Type: preprint
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 $L^2$ 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 $1/2$.
Source: arXiv (Cornell University)
Authors: Han, Bang-Xian, Zhu, Zhuo-Nan
Institution: University of Science and Technology of China
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G2940686378, G6098216124
OpenAlex ID: W7215076832