This question was originaly posted in the stackexchange <https://math.stackexchange.com/questions/1226701/a-measure-on-the-space-of-probability-measures> but since it only got a comment I decided to post it here. I don't know if this is allow, please let me know if it's not. anyway... 


I've been reading about optimal transport and it's connections to geometry. At some point one has to study a bit of the structure of the space of probability measures, $\mathcal{P}(X)$, (over a metric space X) given the 2-Wasserstein metric $W\_2$. I was wondering if there is a 'canonical' way of endowing ($\mathcal{P}(X),W\_2)$ with a 'nice' probability measure. 


More specificaly, let $(X,d)$ be a metric space, and $(\mathcal{P}(X),W\_2)$ it's space of probability measures with the 2-Wasserstein metric. Then, 


1. Can we set a 'nice' measure, $\mu$, on $(\mathcal{P}(X),W\_2)$? (here by nice I guess I mean a non trivial measure that will maybe let us study $(\mathcal{P}(X),W\_2,\mu)$ as a metric mesure space. I realize this is vague, a little guidance here would be appreciated)
2. If we can, what conditions on $X$ are required?
3. Is there any other measure that is usually given to $(\mathcal{P}(X),d)$? Where $d$ can be another metric distinct from $W\_2$.


Thanks, for the time. Any comments and references are highly appreciated!



A measure on the space of probability measures
von Renesse and Sturm have constructed a family "natural" measures $\mu$ making $(\mathcal{P}([0,1]),d\_{W\_2},\mu\_\beta)$ into a metric measure space: <http://www.ams.org/mathscinet-getitem?mr=2537551>. They argue that their measure can be *formally* thought of as 
$$
\tag{\*} \mu\_\beta = C\_\beta^{-1} e^{-\beta Ent(\cdot | \text{Leb})} \mu\_{unif}
$$
where $C\_\beta$ is a normalization constant, $Ent(\rho \text{Leb} |\text{Leb}) = \int\_{[0,1]}\rho \log \rho$ (and is infinite for non-absolutely continuous measures), and $\mu\_{unif}$ is a sort of "uniform" measure. 


Sturm later extended this to allow the underling space to be any $n$-dimensional Riemannian manifold: <http://www.ams.org/mathscinet-getitem?mr=2857031>. 


However, (\*) is not completely rigorous, and the measures do not behave quite as one would expect. There is a formal infinite dimensional Riemannian structure on the regular part of $(\mathcal{P}(M),d\_{W\_2})$ (i.e. on the subset of measures of the form $\rho dV\_g$ where $\rho \in C^\infty(M)$) that goes back to Otto (<http://www.ams.org/mathscinet-getitem?mr=1842429>). One might hope that this Riemannian structure (formally) induced some sort of measure, and that one could use this formal argument to find an actual measure on $\mathcal{P}(M)$, which behaves nicely. However, it seems like while a good notion of "gradient" exists in this formal setting, a Laplacian does not. This suggests that there is no natural measure because otherwise the Laplacian could be defined by integration by parts. See Gigli's monograph <http://www.ams.org/mathscinet-getitem?mr=2920736> (remark 5.6). 


An obvious question given a measure on $(\mathcal{P}(M),d\_{W\_2})$ is whether or not it has lower Ricci curvature bounds in the sense of Lott--Villani--Sturm. When $M$ is the interval, this is not true: <http://www.ams.org/mathscinet-getitem?mr=2900478>. See also Section 8.3 in <http://arxiv.org/pdf/1105.2883.pdf>


In a slightly different direction. Erbar and Huesmann have recently studied <http://link.springer.com/article/10.1007/s00526-014-0790-1> the curvature of the "configuration space" of a manifold and shown that there, there is a natural choice of measure which inherits lower Ricci curvature bounds from the underlying manifold. The configuration space is quite similar to the space of all probability measures (for example, the set of finite sums of dirac measures is dense in the space of probability measures with respect to weak convergence).


