Let $(\mu_t)_{t\in[0,1]}$ be an absolutely continuous curve in the [metric space](/page/Metric%20Space) $(\mathcal P_2(\mathbb R^n),W_2)$, and let $|\mu'|:[0,1]\to[0,\infty)$ denote its metric speed. Then there exists a Borel vector field $v:[0,1]\times\mathbb R^n\to\mathbb R^n$ such that, writing $v_t:=v(t,\cdot)$, the following hold.

For $\mathcal L^1$-a.e. $t\in[0,1]$, one has $v_t\in L^2(\mu_t;\mathbb R^n)$ and

\begin{align*} v_t\in \overline{\{\nabla\phi:\phi\in C_c^\infty(\mathbb R^n)\}}^{L^2(\mu_t;\mathbb R^n)}. \end{align*}

The pair $(\mu_t,v_t)$ solves the continuity equation in the sense that, for every [test function](/page/Test%20Function) $\zeta\in C_c^\infty((0,1)\times\mathbb R^n)$,

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

Moreover, for $\mathcal L^1$-a.e. $t\in[0,1]$,

\begin{align*} \int_{\mathbb R^n}|v_t(x)|^2\,d\mu_t(x)=|\mu'|^2(t). \end{align*}

Finally, if $w:[0,1]\times\mathbb R^n\to\mathbb R^n$ is any Borel vector field such that $w_t\in L^2(\mu_t;\mathbb R^n)$ for $\mathcal L^1$-a.e. $t$,

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

and $(\mu_t,w_t)$ solves the same continuity equation, then for $\mathcal L^1$-a.e. $t\in[0,1]$,

\begin{align*} \int_{\mathbb R^n}|v_t(x)|^2\,d\mu_t(x)\le \int_{\mathbb R^n}|w_t(x)|^2\,d\mu_t(x). \end{align*}