Abstract:We establish several quantitative stability estimates for optimal transport maps between non-degenerate densities on uniformly convex domains for the quadratic cost. Under Hölder regularity assumptions, we prove Lipschitz $L^2$ (respectively $C^{1,\alpha}$) stability estimates for optimal transport maps in terms of the 2-Monge-Kantorovich distance (respectively $L^{p}$ distances) between pairs of source and target densities. When the continuity assumption is removed, we obtain a Lipschitz $L^2$ stability estimate for the Brenier potentials in terms of the $L^2$ distance between the source and target densities.
The proofs rely on a precise characterization of the linear response of the Brenier potential along smooth interpolations of the data, obtained by linearizing the Monge-Ampère equation in divergence form. As a further application of this approach, we derive an explicit formula for the second variation of the quadratic Monge-Kantorovich distance.
| Subjects: | Analysis of PDEs (math.AP) |
| MSC classes: | 49Q22, 35J96, 49K40 |
| Cite as: | arXiv:2605.24232 [math.AP] |
| (or arXiv:2605.24232v1 [math.AP] for this version) | |
| https://doi.org/10.48550/arXiv.2605.24232 arXiv-issued DOI via DataCite (pending registration) |
From: F.U. Caja-Lopez [view email]
[v1]
Fri, 22 May 2026 21:22:07 UTC (41 KB)
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