Let $n \in \mathbb N$, let $T>0$, and let $(\tau_j)_{j\in\mathbb N}$ be a sequence in $(0,\infty)$ such that $\tau_j \to 0$. Let $\mathcal P_2(\mathbb R^n)$ denote the set of Borel probability measures $\mu$ on $\mathbb R^n$ such that

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

Let $W_2$ denote the quadratic Wasserstein distance on $\mathcal P_2(\mathbb R^n)$ induced by the Euclidean distance on $\mathbb R^n$. For each $j\in\mathbb N$, define $N_j:=\lceil T/\tau_j\rceil$, and let $(\rho_{j,k})_{k=0}^{N_j}\subset\mathcal P_2(\mathbb R^n)$. Define the right-continuous piecewise-constant interpolation $\bar\rho_j:[0,T]\to\mathcal P_2(\mathbb R^n)$ by

\begin{align*} \bar\rho_j(0):=\rho_{j,0} \end{align*}

and, for $1\le k\le N_j$, by

\begin{align*} \bar\rho_j(t):=\rho_{j,k} \qquad \text{for } t\in ((k-1)\tau_j,k\tau_j]\cap[0,T]. \end{align*}

Assume that there exist constants $M_T<\infty$ and $C_T<\infty$ such that, for every $j\in\mathbb N$,

\begin{align*} \max_{0\le k\le N_j}\int_{\mathbb R^n}|x|^2\,d\rho_{j,k}(x)\le M_T \end{align*}

and

\begin{align*} \sum_{k=0}^{N_j-1}\frac{W_2^2(\rho_{j,k+1},\rho_{j,k})}{\tau_j}\le C_T. \end{align*}

Then there exist a strictly increasing sequence $(j_m)_{m\in\mathbb N}$ in $\mathbb N$ and a curve $\rho:[0,T]\to\mathcal P_2(\mathbb R^n)$ that is absolutely continuous of order two with respect to $W_2$ such that $\bar\rho_{j_m}(t)$ converges narrowly to $\rho_t$ for every $t\in[0,T]$. Here narrow convergence means convergence against every bounded continuous test function $\mathbb R^n\to\mathbb R$. Absolute continuity of order two means that there exists a function $g\in L^2((0,T);\mathcal L^1)$ such that, for all $0\le s\le t\le T$,

\begin{align*} W_2(\rho_s,\rho_t)\le \int_s^t g(r)\,d\mathcal L^1(r). \end{align*}

The metric derivative $|\rho'|:(0,T)\to[0,\infty]$ is the minimal such speed, equivalently the a.e. limit of $W_2(\rho_{t+h},\rho_t)/|h|$ as $h\to0$ when this limit exists. Moreover $|\rho'|\in L^2((0,T);\mathcal L^1)$ and

\begin{align*} \int_0^T |\rho'|^2(t)\,d\mathcal L^1(t)\le C_T. \end{align*}