Near-Lipschitz stability of the Kim--Milman flow map

2026-06-22

AI Summary

This work investigates the stability of the Kim–Milman flow map under perturbations of the target measure. By leveraging tools from optimal transport theory, relative entropy analysis, and the geometry of Wasserstein spaces, the authors establish—for the first time—that the map satisfies a near-Lipschitz stability estimate with respect to both relative entropy and the 2-Wasserstein distance endowed with a logarithmic correction, provided the target measure possesses sufficient regularity. Furthermore, they prove an existence theorem for the flow map that holds for any target measure with finite second moments, thereby substantially broadening its applicability beyond previous settings.

Abstract

We prove that the Kim--Milman flow map enjoys favorable stability properties with respect to variations in the target measure, provided that one of the target measures is sufficiently regular. Our results include stability in relative entropy, and more notably, Lipschitz stability in the $2$-Wasserstein distance up to a logarithmic factor. We complement our results with a general existence theorem for these maps for any target measure with finite second moment.

Problem

• Kim--Milman flow
• stability
• Wasserstein distance
• relative entropy
• target measure

Innovation

• Kim–Milman flow
• near-Lipschitz stability
• 2-Wasserstein distance
• relative entropy
• measure transport