Suppose that X is the subspace of the set of probability measures on the classical Wiener space $C[0,T]$, for some $T>0$, comprised of Gaussian measures.

In the finite-dimensional setting, the Wasserstein metric between two Gaussian random-variables has a very convenient form.... Are there any known analogues for the infinite dimensional setting? Especially, in the case of the classical Wiener space?

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    $\begingroup$ This is discussed thoroughly in onlinelibrary.wiley.com/doi/abs/10.1002/mana.19901470121. See Section 3 and specifically Theorem 3.5 $\endgroup$ – Nawaf Bou-Rabee Feb 20 '19 at 17:04
  • $\begingroup$ Perfect, I read through most of the paper last night, it's exactly what I'm looking for. If you post this exact comment as an answer, I'll accept it :) $\endgroup$ – N00ber Feb 21 '19 at 11:05

This is discussed thoroughly in the following reference; see Section 3 and specifically Theorem 3.5.

Gelbrich, Matthias, On a formula for the $L^2$ Wasserstein metric between measures on Euclidean and Hilbert spaces, Math. Nachr. 147, 185-203 (1990). ZBL0711.60003.

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    $\begingroup$ Awesome! Thanks Nawaf, great answers as always :) $\endgroup$ – N00ber Feb 21 '19 at 22:02

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