Quantitative stability in optimal transport for general power costs

¹ DMA, Ecole Normale Supérieure PSL, 45 rue d’Ulm, 75005 Paris 5, France
² Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa (PI) Italy

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 13 August 2024
Accepted: 22 April 2026

Abstract

We establish novel quantitative stability results for optimal transport problems with respect to perturbations in the target measure. We provide bounds on the stability of optimal transport potentials and maps, which are relevant for both theoretical and practical applications. Compared to previous results, ours apply to a wide range of costs, including all Wasserstein distances with power cost exponent strictly larger than 1 and leverage mostly assumptions on the source measure, such as log-concavity and bounded support. Our proofs follow the same strategy as [Delalande, A., Quant. Stab. in Quad. Opt. Trans., PhD thesis, 2022] up to several technical improvements. Our work provides a significant step forward in the understanding of stability of optimal transport problems, as previous results were mostly limited to the case of the quadratic cost.