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In Chapter 8 of the book Gradient Flows In Metric Spaces and in the Space of Probability Measures by Ambrosio et al., the tangent space to the space of distributions on $X$ (let's say $X=\mathbb{R}^d$) with finite second moment $\mathscr{P}_2(X)$ is formally defined as $$\text{Tan}_\mu\mathscr{P}_2(X):= \overline{\{\nabla\varphi: \varphi\in C_c^\infty(X)\}}^{L^2(\mu,X)}$$ that is the $L^2$-closure of smooth gradient fields on $X$. I understand the equivalence between absolutely continuous curves on $\mathscr{P}_2(X)$ and minimal norm vector fields, which tells us that the above definition generalizes the construction of the tangent space in finite-dimensional manifolds as classes of differentiable curves.

However, I am curious about how well $\text{Tan}_\mu\mathscr{P}_2(X)$ can serve as a local model for $\mathscr{P}_2(X)$ near $\mu$. In finite-dimensional manifolds, the exponential map gives a local diffeomorphism between the tangent space and the underlying space. Is there a similar construction for $\mathscr{P}_2(X)$? I have read that $\mathscr{P}_2(X)$ is not even a Banach manifold and the above association is merely formal, so the answer is probably no, but I am not sure how severe the failure is.

This paper by J. Lott seems to suggest that restricting to the space $\mathscr{P}^\infty(X)$ of smooth positive densities turns the space into a true manifold and rigorously justifies many geometric concepts. Is there an exponential map in this case, for example?

For motivation, I am trying to prove a version of the center-stable manifold theorem for Wasserstein gradient flows (either weak or strong solutions, with any required degree of regularity). There is already a result for Banach spaces, but to extend the result I need to be able to lift local dynamics on $\mathscr{P}_2(X)$ to the tangent space in some sense.

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    $\begingroup$ I am not entirely sure that I understand your aims, my apologies if my answer is not helpful. I would like to refer you to "An invitation to statistics in the Wasserstein space" by Panaretos and Zemel, Section 2.3 for the exponential maps. On the question regarding the tangent space serving as a local model, arxiv.org/pdf/1910.05954.pdf might be going in the right direction. $\endgroup$
    – Gilles Mordant
    Commented Jan 18 at 12:41
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    $\begingroup$ @GillesMordant Section 2.3 looks exactly like what I'm looking for, but I don't quite understand their approach. In section 2.3.1 it looks like they proved Exp and Log are global bijections between $\mathscr{P}_2(X)$ and $\text{Tan}_\mu\mathscr{P}_2(X)$ (when $\mu$ is absolutely continuous) which doesn't make sense. $\endgroup$
    – Juno Kim
    Commented Jan 18 at 13:42
  • $\begingroup$ What does not make sense exactly? Can you elaborate a bit on what your thoughts are? $\endgroup$
    – Gilles Mordant
    Commented Jan 20 at 19:05
  • $\begingroup$ @GillesMordant A global bijection would imply that $\mathscr{P}_2(X)$ is always isomorphic to a Banach space which is not true. I have been looking at the appendix (Ch.12) of Ambrosio's book as well and I think I understand the issue: Log is not globally well-defined, it's a multivalued operator since transport plans are generally not unique, so Exp cannot give a local model for $\mathscr{P}_2(X)$ unless we restrict to smooth densities. $\endgroup$
    – Juno Kim
    Commented Jan 21 at 3:43
  • $\begingroup$ In that field, people care about the tangent space at one fixed, smooth measure, which might explain the differing views. $\endgroup$
    – Gilles Mordant
    Commented Jan 22 at 9:52

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