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GT EYAWKAJKOS
Quantitative Stability, Coercivity and Uniqueness of Optimal Transport Plans
par
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Europe/Paris
112 (Braconnier)
112
Braconnier
Description
We prove three quantitative results regarding optimal transport plans for the quadratic cost:
1. Bi-marginal quantitative stability: For probability measures on R^d supported in fixed compact sets, we prove that quadratic optimal transport plans are quantitatively stable in Wasserstein distance under perturbation of both marginal measures, assuming that one of the initial measures satisfies an upper Ahlfors regularity condition with exponent strictly greater than d-1.
2. Coercivity: Under the same Ahlfors assumption, we prove that any transport plan on the product space must be quantitatively close to the optimal plan, if it has similar marginals and a similar transport cost to the optimal value.
3. Quantitative Uniqueness: We prove a quantitative version of Brenier's theorem. The Wasserstein diameter of the set of optimal plans is controlled by the Wasserstein distance of one marginal measure to a regular measure for which uniqueness holds. In this way, "almost uniqueness'' of optimal plans is quantified by the source measure being "almost regular''.
2. Coercivity: Under the same Ahlfors assumption, we prove that any transport plan on the product space must be quantitatively close to the optimal plan, if it has similar marginals and a similar transport cost to the optimal value.
3. Quantitative Uniqueness: We prove a quantitative version of Brenier's theorem. The Wasserstein diameter of the set of optimal plans is controlled by the Wasserstein distance of one marginal measure to a regular measure for which uniqueness holds. In this way, "almost uniqueness'' of optimal plans is quantified by the source measure being "almost regular''.