Let $n\in\mathbb N$ with $n\ge 1$, let $\varepsilon>0$, and set $I:=(-\varepsilon,\varepsilon)$. Let $\rho:I\to \mathcal P_2(\mathbb R^n)$, $t\mapsto\rho_t$, be a $2$-absolutely continuous curve in the metric space $(\mathcal P_2(\mathbb R^n),W_2)$. Assume that, for every $t\in I$, there exists a density $r_t\in C^\infty(\mathbb R^n;(0,\infty))$ such that $\rho_t=r_t\mathcal L^n$ and

\begin{align*} \int_{\mathbb R^n}|x|^2r_t(x)\,d\mathcal L^n(x)<\infty. \end{align*}

Assume also that $t\mapsto r_t$ is differentiable as a distribution-valued map for $\mathcal L^1$-almost every $t\in I$, and denote its derivative at such a time by $\partial_t r_t\in\mathcal D'(\mathbb R^n)$. For every such time $t$, define

\begin{align*} \mathcal A_t:=\left\{v\in L^2(\rho_t;\mathbb R^n): (\partial_t r_t)(\psi)=\int_{\mathbb R^n}\nabla\psi(x)\cdot v(x)r_t(x)\,d\mathcal L^n(x)\text{ for every }\psi\in C_c^\infty(\mathbb R^n)\right\}. \end{align*}

Then, for $\mathcal L^1$-almost every $t\in I$,

\begin{align*} |\dot\rho_t|_{W_2}^2=\inf_{v\in\mathcal A_t}\int_{\mathbb R^n}|v(x)|^2\,d\rho_t(x). \end{align*}

Moreover, suppose that for such a time $t$ the infimum is attained by a vector field $v\in C^\infty(\mathbb R^n;\mathbb R^n)\cap L^2(\rho_t;\mathbb R^n)$. Then there exists a smooth function $\phi:\mathbb R^n\to\mathbb R$ such that $v=\nabla\phi$, and therefore

\begin{align*} |\dot\rho_t|_{W_2}^2=\int_{\mathbb R^n}|\nabla\phi(x)|^2r_t(x)\,d\mathcal L^n(x). \end{align*}