Benjamin Jourdain - Regularity of the quadratic Wasserstein projections in the convex order скачать в хорошем качестве

Benjamin Jourdain - Regularity of the quadratic Wasserstein projections in the convex order

This lecture was part of the Workshop on "Probabilistic Mass Transport - from Schrödinger to Stochastic Analysis" held at the ESI February 9 - 13, 2026. We show continuity of both Wasserstein projections in the convex order when they are unique. We also check that, in arbitrary dimension d, the quadratic Wasserstein projection of a probability measure μ on the set of probability measures dominated by ν in the convex order is non-expansive in μ and Hölder continuous with exponent 1/2 in ν. When μ and ν are Gaussian, we check that this projection is Gaussian and also consider the quadratic Wasserstein projection on the set of probability measures ν dominating μ in the convex order. In the case when d≥2 and ν is not absolutely continuous with respect to the Lebesgue measure where uniqueness of the latter projection was not known, we check that there is always a unique Gaussian projection and characterize when non Gaussian projections with the same covariance matrix also exist. Still for Gaussian distributions, we characterize the covariance matrices of the two projections. It turns out that there exists an orthogonal transformation of space under which the computations are similar to the easy case when the covariance matrices of μ and ν are diagonal.