First variation in $L^2$ sense of the square of the Wasserstein metric

Question

Let me consider the functional $\mathcal{F}(\rho)=\mathcal{W}_2^2(\mu,\rho)$ defined in in the space of absolutely continuous probability measures $\mathcal{P}_{ac,2}(\Omega)$, where $\mathcal{W}_2^2$ stand for the Wasserstein metric.

If $\Omega$ is a bounded domain, it is well known that the first variation, in $L^2$ sense, of $\mathcal{F}$ at the point $\rho$ is given by the Kantorovich potential betwen $\mu$ and $\rho$. The proof essentially uses a stability result about the Kantorovich potentials which is a consequence of Ascoli-Arzela's Theorem. Details can be founded in the book of Santambrogio "Optimal transport for aplied mathematicians" chapter 7.

My question is: If $\Omega=\mathbb{R}^d$, this result holds true? how the proof can be adapted? I tried to show it but I feel that some additional restrictive conditions must be added.

Thanks.

Answer 1

The question of differentiability of Wasserstein distance is addressed in this paper https://arxiv.org/pdf/1811.07787.pdf. They show that under certain conditions, the map $ \sigma \mapsto W(\sigma,\nu)$ is differentiable at $\mu$ in Lions' sense.