Let $n\in\mathbb N$, and let $\mathcal P_2(\mathbb R^n)$ denote the set of Borel probability measures on $\mathbb R^n$ with finite second moment, equipped with the quadratic Wasserstein distance $W_2$.

If $\mu:[0,1]\to\mathcal P_2(\mathbb R^n)$, $t\mapsto\mu_t$, is $2$-absolutely continuous as a curve in the metric space $(\mathcal P_2(\mathbb R^n),W_2)$, meaning that its metric derivative $|\mu'|$ exists for $\mathcal L^1$-a.e. $t\in(0,1)$ and belongs to $L^2((0,1),\mathcal B((0,1)),\mathcal L^1)$, then there exists a Borel map $v:(0,1)\times\mathbb R^n\to\mathbb R^n$ such that, for $\mathcal L^1$-a.e. $t\in(0,1)$, the section $v_t:\mathbb R^n\to\mathbb R^n$ defined by $v_t(x):=v(t,x)$ belongs to $L^2(\mathbb R^n,\mathcal B(\mathbb R^n),\mu_t;\mathbb R^n)$, the pair $(\mu_t,v_t)$ solves the continuity equation

\begin{align*} \int_0^1\int_{\mathbb R^n}\partial_t\varphi(t,x)\,d\mu_t(x)\,d\mathcal L^1(t)+\int_0^1\int_{\mathbb R^n}\nabla_x\varphi(t,x)\cdot v_t(x)\,d\mu_t(x)\,d\mathcal L^1(t)=0 \end{align*}

for every $\varphi\in C_c^\infty((0,1)\times\mathbb R^n)$, and

\begin{align*} \left(\int_{\mathbb R^n}|v_t(x)|^2\,d\mu_t(x)\right)^{1/2}\le |\mu'|(t) \end{align*}

for $\mathcal L^1$-a.e. $t\in(0,1)$.

Conversely, suppose that $\mu:[0,1]\to\mathcal P_2(\mathbb R^n)$, $t\mapsto\mu_t$, is narrowly continuous and that there exists a Borel map $v:(0,1)\times\mathbb R^n\to\mathbb R^n$ such that $v_t\in L^2(\mathbb R^n,\mathcal B(\mathbb R^n),\mu_t;\mathbb R^n)$ for $\mathcal L^1$-a.e. $t\in(0,1)$,

\begin{align*} \int_0^1\int_{\mathbb R^n}|v_t(x)|^2\,d\mu_t(x)\,d\mathcal L^1(t)<\infty, \end{align*}

and $(\mu_t,v_t)$ solves the continuity equation in the distributional sense above. Then $\mu$ is $2$-absolutely continuous as a curve in $(\mathcal P_2(\mathbb R^n),W_2)$ and

\begin{align*} |\mu'|(t)\le \left(\int_{\mathbb R^n}|v_t(x)|^2\,d\mu_t(x)\right)^{1/2} \end{align*}

for $\mathcal L^1$-a.e. $t\in(0,1)$.