Perturbation of Wasserstein distance: looking for references

Question

I am doing readings related to Optimal transport which is new to me and I often encounter the following statement regarding a sort of derivative of the Wasserstein distance: $u$ and $v$ be two probability densities on $\Bbb R^d$ with finite second moment, let $\eta\in C_c^\infty(\Bbb R^d,\Bbb R^d)$, define $$\phi_{\delta}(x) := x + \delta \eta(x)$$ consider the inner perturbation $$u_\delta(x) = (\phi_\delta)_\# u= \det(D \phi_\delta)^{-1} u\circ \phi_\delta^{-1}$$

Then it appears that the following holds

\begin{align*} \lim_{\delta \to 0} \frac{1}{2\delta} \left[ W^{2}(u_{\delta}, v) - W^{2}(u,v) \right] = \int_{\Bbb R^{d}}\big(T_{u}^{v} - \mathrm{I}\big) \cdot \eta u \, d x, \end{align*} here $T_{u}^{v}$ is the optimal transport map pushing $u$ to $v$, i.e. $(T_{u}^{v})_\#u=v$ and $W$ is the natural Wasserstein distance in $\mathcal{P}_2(\Bbb R^d)$

Question where can I find a reference to this result?

Answer 1

You can find this in Villani's "small book", Theorem 8.13 in [Villani, C. (2003). Topics in optimal transportation (Vol. 58). American Mathematical Soc.]

I can also recommend looking at Filippo Santanbrogio's more recent and applied book [Santambrogio, F. (2015). Optimal transport for applied mathematicians. Birkäuser, NY, 55(58-63), 94.] in particular chapter 7 (a more general version of your statement can be found in section 7.2.2)