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While working on my thesis, I encountered the idea of OMT and started reading some more (like Villani's book). In particular, I came across a PhD thesis by Martial Agueh. I thought it was interesting because it explicitly investigated the geodesics of Wasserstein space to produce solutions to a type of parabolic PDE.

I have some questions and I know this is probably something I should reach out to a researcher about, but I'm in another very different field (computational neuroscience) and few people around me that have the ability to address these questions with sufficient detail -- so I figure I'd try my luck and ask here.

  1. Are there any open questions that would require a detailed study of the geodesics in Wasserstein space to understand the existence of PDE solutions (as in Agueh's thesis) or are these results mostly covered now by the general theory in Ambrosio's book on Gradient Flows?

  2. Are there any open questions about analysis on Wasserstein space itself? I think one game people try to play is defining Laplacians on this space. Anything else?

  3. Are there any PDE problems that have not yet been looked at through the Wasserstein perspective?

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    $\begingroup$ Please read the tag descriptions before using a tag. The [open-problems] tag is for indicating that the mathematical question being asked turns out to be equivalent to a known open problem. $\endgroup$
    – Willie Wong
    Commented Mar 20, 2019 at 14:22

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I work in a bit different field too, but your question is quite interesting for me. So I have looked up in the literature just for curiosity. Since there are (surprisingly) no answers yet, let me share some references where open problems related to Wasserstein space are mentioned:

  1. Topics in Optimal Transportation by C. Villani (2003).
    For instance see Open Problem 7.20.

  2. A geometric study of Wasserstein spaces: Euclidean spaces by B. Kloeckner, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Volume 9 (2010) no. 2, p. 297-323.

  3. A user’s guide to optimal transport by L. Ambrosio and N. Gigli (2012).
    For instance see Open Problem 5.7.

  4. { Euclidean, Metric, and Wasserstein } Gradient Flows: an overview by F. Santambrogio (2016).

In addition, I am not aware if an explicit formula for $W_p$ distance between two Gaussian measures is known for $p\ne 2$, see e.g. this question.

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