# Reference Request: 2-Wasserstein Metric on Wiener Space

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?

• This is discussed thoroughly in onlinelibrary.wiley.com/doi/abs/10.1002/mana.19901470121. See Section 3 and specifically Theorem 3.5 – Nawaf Bou-Rabee Feb 20 '19 at 17:04
• 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 :) – N00ber Feb 21 '19 at 11:05

## 1 Answer

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.

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