On a Formula for the L 2 Wasserstein Metric between Measures on Euclidean and Hilbert Spaces

For a separable metric space ( X, d ) L p Wasserstein metrics between probability measures μ and v on X are defined by\documentclass{article}\pagestyle{empty}\begin{document}$$ W_p \left({\mu,\nu } \right): = \inf \left\{ {\left({\mathop f\limits_{x \times x} d^p \left({x,y} \right)d\eta } \right)^{1/p} } \right\},p\;\varepsilon \;\left({1,\infty } \right),$$\end{document}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.