One great inside of Felix Otto is that the Wasserstein metric from optimal transportation gives the space of (finite second moment, probability) measures on $\mathbb{R}^n$ (or a manifold) a kind of differentiable structure. This insight has been used a lot for gradient flows, using mostly differentiability almost everywhere (e.g. for absolutely continuous curves).

Nicola Gigli has set up a framework that gives sense to the differential at a given point of a map taking measures as argument. I know very few uses of this concept, but have the feeling it could be a great application of optimal transport.

So here is the question:

What are examples of possible use of classical differential calculus tools in a space of probability measures ?

Slightly more precise versions of the question can be:

Q1:what are examples of family of functionals, defined on a space of measure, with meaningful zeros, for which an implicit function theorem would be useful?

Q2:what are examples of maps on a space of measure that we would want to inverse locally?

Let me give an example for illustration. The map $\times 2 : x\mapsto 2x \mod 1$ acts on the circle, and by push-forward (denoted by $\times 2_\#$) it acts on probability measures on the circle. The Lebesgue measure is invariant, so it is a fixed point of $\times 2_\#$. One can prove that in Gigli's framework, this map is differentiable at Lebesgue measure and the differential can be computed explicitly. The spectral properties are quite ugly, as every complex number in the open disc of radius $2$ is an eigenvalue of infinite multiplicity. This has fun consequences, but is quite far from usual hypotheses of differential calculus theorems.