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Abstract For a separable metric space ( X, d ) L p Wasserstein metrics between probability measures μ and v on X are defined by where the infimum is taken over all probability measures η on X × X with marginal distributions μ and v , respectively. After mentioning some basic properties of these metrics as well as explicit formulae for X = R a formula for the L 2 Wasserstein metric with X = R n will be cited from 5, 9, and 21 and proved for any two probability measures of a family of elliptically contoured distributions. Finally this result will be generalized for Gaussian measures to the case of a separable Hilbert space.
Matthias Gelbrich (Mon,) studied this question.