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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.

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Just a remark: the example you gave is in fact standard in the theory of discrete dynamical systems; note that your operator is already a bounded linear operator on the space of Radon measures. There is a lot of material about the spectral properties of push-forward operators in D.S.: maybe you can find there other examples/suggestions. –  Pietro Majer Jul 3 '13 at 12:13
@Pietro Majer: I do not think that the given example is standard; beware that I do mean the push-forward acting on all measures, not only absolutely continuous ones (in particular, this is not a linear action on a function space). Notably, the derivative at Lebesgue measure is not the transfer operator (it is close to it for $\times 2$, but quite different for other expanding circle maps). Finally, I have to precise that the differential structure that comes from optimal transport is very different from the one coming from the affine structure. –  Benoît Kloeckner Jul 3 '13 at 14:03
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