Regularity of the Quadratic Wasserstein Projections in the Convex Order

Explore the mathematical properties of quadratic Wasserstein projections in convex order through this 34-minute conference lecture from the Erwin Schrödinger International Institute for Mathematics and Physics. Examine the continuity conditions for Wasserstein projections when uniqueness is established, and discover how these projections behave as non-expansive operators in μ and exhibit Hölder continuity with exponent 1/2 in ν across arbitrary dimensions. Investigate the special case of Gaussian probability measures, where projections maintain their Gaussian nature, and learn about the characterization of covariance matrices for both types of projections. Delve into the complex scenario where d≥2 and ν lacks absolute continuity with respect to Lebesgue measure, understanding when unique Gaussian projections exist and when multiple non-Gaussian projections with identical covariance matrices can occur. Master the orthogonal transformation techniques that simplify computations to cases analogous to diagonal covariance matrices, providing powerful tools for analyzing probabilistic mass transport problems in stochastic analysis.