[proofplan]
We invoke the standard Ambrosio-Gigli-Savare tangent-velocity theorem for absolutely continuous curves in $\mathcal P_2(\mathbb R^n)$ in the precise form needed here: it gives a Borel velocity field in the closed gradient tangent space, the continuity equation, and the bound by the metric speed. The reverse inequality between metric speed and any continuity-equation velocity gives equality of the norm with $|\mu'|$. The final minimality statement follows from the Hilbert-space [orthogonal projection](/theorems/437) of an arbitrary representing velocity onto the closed gradient subspace.
[/proofplan]
[step:Invoke the external tangent-velocity theorem with verified hypotheses]
For each $t\in[0,1]$, define the closed gradient subspace $G_t\subset L^2(\mu_t;\mathbb R^n)$ by
\begin{align*}
G_t:=\overline{\{\nabla\phi:\phi\in C_c^\infty(\mathbb R^n)\}}^{L^2(\mu_t;\mathbb R^n)}.
\end{align*}
We use the following external standard theorem, usually stated as the tangent-space characterization of absolutely continuous curves in Wasserstein space.
External standard theorem: if $(\nu_t)_{t\in[0,1]}$ is absolutely continuous in $(\mathcal P_2(\mathbb R^n),W_2)$, then $t\mapsto\nu_t$ is narrowly continuous, its metric speed $|\nu'|$ belongs to $L^1(0,1)$, and there exists a Borel map $u:[0,1]\times\mathbb R^n\to\mathbb R^n$ such that $u_t\in L^2(\nu_t;\mathbb R^n)$ and $u_t\in\overline{\{\nabla\phi:\phi\in C_c^\infty(\mathbb R^n)\}}^{L^2(\nu_t;\mathbb R^n)}$ for $\mathcal L^1$-a.e. $t$, the pair $(\nu_t,u_t)$ solves the distributional continuity equation, and
\begin{align*}
\|u_t\|_{L^2(\nu_t;\mathbb R^n)}\le |\nu'|(t)
\end{align*}
for $\mathcal L^1$-a.e. $t$. The same theorem includes the measurable-selection assertion: the representatives may be chosen as a jointly Borel map on $[0,1]\times\mathbb R^n$. It also includes the converse metric-speed estimate: whenever a Borel field $a:[0,1]\times\mathbb R^n\to\mathbb R^n$ satisfies $a_t\in L^2(\nu_t;\mathbb R^n)$ for a.e. $t$, solves the continuity equation with $(\nu_t)$, and has locally integrable $L^2(\nu_t)$ norm in time, then
\begin{align*}
|\nu'|(t)\le \|a_t\|_{L^2(\nu_t;\mathbb R^n)}
\end{align*}
for $\mathcal L^1$-a.e. $t$.
The hypotheses of this external theorem are satisfied with $\nu_t:=\mu_t$ because the theorem assumes exactly that $(\mu_t)_{t\in[0,1]}$ is absolutely continuous in $(\mathcal P_2(\mathbb R^n),W_2)$. Applying it gives a Borel vector field $v:[0,1]\times\mathbb R^n\to\mathbb R^n$ with $v_t\in G_t$ for $\mathcal L^1$-a.e. $t$, with $(\mu_t,v_t)$ solving the displayed continuity equation, and with
\begin{align*}
\|v_t\|_{L^2(\mu_t;\mathbb R^n)}\le |\mu'|(t)
\end{align*}
for $\mathcal L^1$-a.e. $t$.
[guided]
The nontrivial analytic input is not a pointwise Riesz construction made independently at each time. Such a construction would leave two serious gaps: simultaneous differentiability for all smooth tests and measurability of representatives as $t$ varies. We therefore use the standard tangent-velocity theorem for Wasserstein absolutely continuous curves in its full measurable form.
For each $t\in[0,1]$, set
\begin{align*}
G_t:=\overline{\{\nabla\phi:\phi\in C_c^\infty(\mathbb R^n)\}}^{L^2(\mu_t;\mathbb R^n)}.
\end{align*}
This is a closed linear subspace of the [Hilbert space](/page/Hilbert%20Space) $L^2(\mu_t;\mathbb R^n)$.
The external tangent-velocity theorem states the following. If $(\nu_t)_{t\in[0,1]}$ is absolutely continuous in $(\mathcal P_2(\mathbb R^n),W_2)$, then the metric derivative $|\nu'|$ belongs to $L^1(0,1)$, the curve is narrowly continuous, and there is a jointly Borel map $u:[0,1]\times\mathbb R^n\to\mathbb R^n$ such that $u_t$ lies in the closed gradient subspace of $L^2(\nu_t;\mathbb R^n)$ for a.e. $t$, the pair $(\nu_t,u_t)$ solves the distributional continuity equation, and
\begin{align*}
\|u_t\|_{L^2(\nu_t;\mathbb R^n)}\le |\nu'|(t)
\end{align*}
for a.e. $t$. The same theorem gives the converse estimate for any Borel velocity field representing the same curve through the continuity equation:
\begin{align*}
|\nu'|(t)\le \|a_t\|_{L^2(\nu_t;\mathbb R^n)}
\end{align*}
for a.e. $t$ whenever the right-hand side is locally integrable in time.
We verify the theorem's hypotheses. In the present theorem, the curve $(\mu_t)_{t\in[0,1]}$ is assumed absolutely continuous in $(\mathcal P_2(\mathbb R^n),W_2)$, and this is exactly the input hypothesis of the external tangent-velocity theorem with $\nu_t:=\mu_t$. Hence the theorem supplies a Borel map $v:[0,1]\times\mathbb R^n\to\mathbb R^n$ satisfying $v_t\in G_t$ for $\mathcal L^1$-a.e. $t$, the distributional continuity equation, and the estimate
\begin{align*}
\|v_t\|_{L^2(\mu_t;\mathbb R^n)}\le |\mu'|(t)
\end{align*}
for $\mathcal L^1$-a.e. $t$.
[/guided]
[/step]
[step:Check that the continuity-equation integrals are finite]
Let $\zeta\in C_c^\infty((0,1)\times\mathbb R^n)$ be a [test function](/page/Test%20Function). Since $\zeta$ is smooth with compact support, the functions $\partial_t\zeta$ and $\nabla_x\zeta$ are bounded. The first spacetime integral satisfies
\begin{align*}
\int_0^1\int_{\mathbb R^n}|\partial_t\zeta(t,x)|\,d\mu_t(x)\,d\mathcal L^1(t)\le \|\partial_t\zeta\|_\infty.
\end{align*}
For the second integral, Cauchy-Schwarz in the Hilbert space $L^2(\mu_t;\mathbb R^n)$ gives
\begin{align*}
\int_{\mathbb R^n}|\nabla_x\zeta(t,x)\cdot v_t(x)|\,d\mu_t(x)\le \|\nabla_x\zeta\|_\infty\|v_t\|_{L^2(\mu_t;\mathbb R^n)}.
\end{align*}
Using the estimate from the preceding step and $|\mu'|\in L^1(0,1)$, we obtain
\begin{align*}
\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)\le \|\nabla_x\zeta\|_\infty\int_0^1|\mu'|(t)\,d\mathcal L^1(t)<\infty.
\end{align*}
Thus the distributional continuity equation supplied by the tangent-velocity theorem is well-defined for every $\zeta\in C_c^\infty((0,1)\times\mathbb R^n)$.
[/step]
[step:Use the converse speed estimate to identify the norm of the constructed field]
The preceding step shows that $v$ is an admissible velocity field for the converse metric-speed estimate in the external tangent-velocity theorem. Therefore
\begin{align*}
|\mu'|(t)\le \|v_t\|_{L^2(\mu_t;\mathbb R^n)}
\end{align*}
for $\mathcal L^1$-a.e. $t$. Combining this with the already established inequality
\begin{align*}
\|v_t\|_{L^2(\mu_t;\mathbb R^n)}\le |\mu'|(t)
\end{align*}
gives
\begin{align*}
\|v_t\|_{L^2(\mu_t;\mathbb R^n)}=|\mu'|(t)
\end{align*}
for $\mathcal L^1$-a.e. $t$. Equivalently,
\begin{align*}
\int_{\mathbb R^n}|v_t(x)|^2\,d\mu_t(x)=|\mu'|^2(t)
\end{align*}
for $\mathcal L^1$-a.e. $t$.
[/step]
[step:Compare any other velocity with the closed-gradient representative]
Let $w:[0,1]\times\mathbb R^n\to\mathbb R^n$ be a Borel vector field satisfying the hypotheses in the theorem. For $\mathcal L^1$-a.e. $t$, both $v_t$ and $w_t$ belong to $L^2(\mu_t;\mathbb R^n)$, and $v_t\in G_t$.
The weak continuity equations for $v$ and $w$ imply that the two fields induce the same infinitesimal action on every smooth spatial test. More precisely, for every $\phi\in C_c^\infty(\mathbb R^n)$, define $K_\phi:[0,1]\to\mathbb R$ by
\begin{align*}
K_\phi(t):=\int_{\mathbb R^n}\phi(x)\,d\mu_t(x).
\end{align*}
Testing the continuity equation for $v$ and for $w$ with products $\zeta(t,x)=\eta(t)\phi(x)$, where $\eta\in C_c^\infty((0,1))$, gives the same [distributional derivative](/page/Distributional%20Derivative) of $K_\phi$. Hence, for $\mathcal L^1$-a.e. $t$,
\begin{align*}
\int_{\mathbb R^n}\nabla\phi(x)\cdot v_t(x)\,d\mu_t(x)=\int_{\mathbb R^n}\nabla\phi(x)\cdot w_t(x)\,d\mu_t(x).
\end{align*}
Using the standard separability argument for $C_c^\infty(\mathbb R^n)$ with the topology of [uniform convergence](/page/Uniform%20Convergence) of functions and first derivatives on compact sets, we choose one full-measure set of times on which this identity holds for a countable dense family of tests and then extend it by continuity of the two $L^2(\mu_t)$ pairings. Therefore, for $\mathcal L^1$-a.e. $t$,
\begin{align*}
\int_{\mathbb R^n}g(x)\cdot (w_t(x)-v_t(x))\,d\mu_t(x)=0
\end{align*}
for every $g\in G_t$.
Since $v_t\in G_t$, we may take $g=v_t$ and obtain
\begin{align*}
\int_{\mathbb R^n}v_t(x)\cdot (w_t(x)-v_t(x))\,d\mu_t(x)=0.
\end{align*}
Expanding the square in the Hilbert space $L^2(\mu_t;\mathbb R^n)$ yields
\begin{align*}
\int_{\mathbb R^n}|w_t(x)|^2\,d\mu_t(x)=\int_{\mathbb R^n}|v_t(x)|^2\,d\mu_t(x)+\int_{\mathbb R^n}|w_t(x)-v_t(x)|^2\,d\mu_t(x).
\end{align*}
The last term is nonnegative, so
\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*}
for $\mathcal L^1$-a.e. $t$. This proves the minimality assertion and completes the proof.
[guided]
We now prove the pointwise minimality statement. The idea is Hilbert-space orthogonality: $v_t$ is the closed-gradient part of any other velocity field representing the same infinitesimal motion, so the remainder $w_t-v_t$ is orthogonal to the closed gradient subspace.
Let $w:[0,1]\times\mathbb R^n\to\mathbb R^n$ be a Borel vector field satisfying the hypotheses in the theorem. Thus $w_t\in L^2(\mu_t;\mathbb R^n)$ for a.e. $t$, the spacetime integral of $|w_t|^2$ is finite, and $(\mu_t,w_t)$ solves the same continuity equation as $(\mu_t,v_t)$.
Fix a spatial test function $\phi\in C_c^\infty(\mathbb R^n)$ and define $K_\phi:[0,1]\to\mathbb R$ by
\begin{align*}
K_\phi(t):=\int_{\mathbb R^n}\phi(x)\,d\mu_t(x).
\end{align*}
For every $\eta\in C_c^\infty((0,1))$, use the product test function $\zeta:[0,1]\times\mathbb R^n\to\mathbb R$ given by $\zeta(t,x):=\eta(t)\phi(x)$. Substituting this test into the continuity equation for $v$ gives the distributional identity
\begin{align*}
\int_0^1 K_\phi(t)\eta'(t)\,d\mathcal L^1(t)+\int_0^1\eta(t)\int_{\mathbb R^n}\nabla\phi(x)\cdot v_t(x)\,d\mu_t(x)\,d\mathcal L^1(t)=0.
\end{align*}
Substituting the same test into the continuity equation for $w$ gives
\begin{align*}
\int_0^1 K_\phi(t)\eta'(t)\,d\mathcal L^1(t)+\int_0^1\eta(t)\int_{\mathbb R^n}\nabla\phi(x)\cdot w_t(x)\,d\mu_t(x)\,d\mathcal L^1(t)=0.
\end{align*}
The first term is identical in the two identities because the measure curve is the same. Subtracting the two identities gives
\begin{align*}
\int_0^1\eta(t)\int_{\mathbb R^n}\nabla\phi(x)\cdot (w_t(x)-v_t(x))\,d\mu_t(x)\,d\mathcal L^1(t)=0.
\end{align*}
Since this holds for every $\eta\in C_c^\infty((0,1))$, the fundamental lemma for distributions gives
\begin{align*}
\int_{\mathbb R^n}\nabla\phi(x)\cdot (w_t(x)-v_t(x))\,d\mu_t(x)=0
\end{align*}
for $\mathcal L^1$-a.e. $t$, for this fixed $\phi$.
To pass from one fixed test to all gradients in $G_t$, we use the standard countable-dense-family argument. Choose a countable family of smooth compactly supported functions whose gradients are dense in the collection of smooth compactly supported gradients on each compact set after taking rational coefficients and rational mollifier parameters. Intersecting the corresponding full-measure sets gives a single full-measure set of times where the orthogonality identity holds for that countable family. For such a time $t$, the maps
\begin{align*}
g\mapsto \int_{\mathbb R^n}g(x)\cdot v_t(x)\,d\mu_t(x)
\end{align*}
and
\begin{align*}
g\mapsto \int_{\mathbb R^n}g(x)\cdot w_t(x)\,d\mu_t(x)
\end{align*}
are continuous linear functionals on $L^2(\mu_t;\mathbb R^n)$ by the [Cauchy-Schwarz inequality](/theorems/432). Therefore the orthogonality identity extends from the dense family to every $g\in G_t$:
\begin{align*}
\int_{\mathbb R^n}g(x)\cdot (w_t(x)-v_t(x))\,d\mu_t(x)=0.
\end{align*}
Now use the fact that $v_t\in G_t$. Taking $g=v_t$ gives
\begin{align*}
\int_{\mathbb R^n}v_t(x)\cdot (w_t(x)-v_t(x))\,d\mu_t(x)=0.
\end{align*}
Thus $v_t$ and $w_t-v_t$ are orthogonal in the Hilbert space $L^2(\mu_t;\mathbb R^n)$. Expanding the square using this orthogonality gives
\begin{align*}
\int_{\mathbb R^n}|w_t(x)|^2\,d\mu_t(x)=\int_{\mathbb R^n}|v_t(x)|^2\,d\mu_t(x)+\int_{\mathbb R^n}|w_t(x)-v_t(x)|^2\,d\mu_t(x).
\end{align*}
The second term on the right is nonnegative. Hence
\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*}
for $\mathcal L^1$-a.e. $t$. This is exactly the asserted minimality of the constructed velocity.
[/guided]
[/step]
