A curve of measures has no pointwise derivative in the usual vector-space sense. Its infinitesimal motion is instead described by a velocity field transporting mass through the continuity equation. Metric absolute continuity and this Eulerian description are equivalent.
The distributional continuity equation
Definition. A narrowly continuous curve \(t\mapsto\mu_t\in\mathcal P(\mathbb R^d)\) and a Borel velocity field \(v_t(x)\) solve \[ \partial_t\mu_t+\nabla\cdot(v_t\mu_t)=0 \] on \((0,T)\) if, for every \(\zeta\in C_c^\infty((0,T)\times\mathbb R^d)\), \[ \int_0^T\int_{\mathbb R^d} \{\partial_t\zeta(t,x)+\nabla\zeta(t,x)\cdot v_t(x)\} \,\mathrm{d}\mu_t(x)\,\mathrm{d}t=0. \]
Equivalently, for every \(\phi\in C_c^\infty(\mathbb R^d)\), the map \(t\mapsto\int\phi\,\mathrm{d}\mu_t\) is absolutely continuous and \[ \frac{\mathrm{d}}{\mathrm{d}t}\int\phi\,\mathrm{d}\mu_t =\int\nabla\phi(x)\cdot v_t(x)\,\mathrm{d}\mu_t(x) \] for almost every \(t\).
Flows of particles
Proposition. Suppose \(v_t\) is smooth and \(X_t(x)\) solves \[ \frac{\mathrm{d}}{\mathrm{d}t}X_t(x)=v_t(X_t(x)),\qquad X_0(x)=x. \] Then \(\mu_t=X_{t\#}\mu_0\) solves the continuity equation.
Proof. For a smooth compactly supported \(\phi\), \[ \frac{\mathrm{d}}{\mathrm{d}t}\int\phi\,\mathrm{d}\mu_t =\frac{\mathrm{d}}{\mathrm{d}t}\int\phi(X_t(x))\,\mathrm{d}\mu_0(x) =\int\nabla\phi(X_t(x))\cdot v_t(X_t(x))\,\mathrm{d}\mu_0(x). \] Pushing forward the last integral gives \(\int\nabla\phi\cdot v_t\,\mathrm{d}\mu_t\), which is the weak equation.
Metric absolute continuity
Definition. A curve \(\mu:[0,T]\to(\mathcal P_p(\mathbb R^d),W_p)\) belongs to \(AC^p\) if there exists \(m\in L^p(0,T)\) such that \[ W_p(\mu_s,\mu_t)\le\int_s^t m(r)\,\mathrm{d}r \] whenever \(s\le t\). Its metric derivative is \[ |\dot\mu_t|=\lim_{h\to0}\frac{W_p(\mu_{t+h},\mu_t)}{|h|}, \] which exists for almost every \(t\), belongs to \(L^p\), and is the smallest admissible \(m\).
Velocity characterization
Eulerian characterization. Let \(1<p<\infty\). A curve \(\mu_t\) belongs to \(AC^p([0,T];\mathcal P_p(\mathbb R^d))\) if and only if there exists a Borel field \(v_t\in L^p(\mu_t;\mathbb R^d)\) solving the continuity equation and satisfying \[ \int_0^T\|v_t\|_{L^p(\mu_t)}^p\,\mathrm{d}t<\infty. \] Every such field obeys \[ |\dot\mu_t|\le\|v_t\|_{L^p(\mu_t)} \quad\text{a.e.} \] There is a unique minimal velocity in the appropriate tangent class for which equality holds.
Proof, velocity implies absolute continuity. The superposition principle represents the solution by a probability \(\boldsymbol\eta\) on absolutely continuous paths \(\gamma\) such that \((e_t)_\#\boldsymbol\eta=\mu_t\) and \(\dot\gamma_t=v_t(\gamma_t)\) almost everywhere. The joint law of \((\gamma_s,\gamma_t)\) couples \(\mu_s,\mu_t\), so \[ \begin{aligned} W_p(\mu_s,\mu_t) &\le\left(\int|\gamma_t-\gamma_s|^p\,\mathrm{d}\boldsymbol\eta\right)^{1/p}\\ &\le\int_s^t \left(\int|\dot\gamma_r|^p\,\mathrm{d}\boldsymbol\eta\right)^{1/p}\mathrm{d}r\\ &=\int_s^t\|v_r\|_{L^p(\mu_r)}\,\mathrm{d}r. \end{aligned} \] This proves absolute continuity and the metric-derivative bound.
Proof, absolute continuity implies velocity. For small \(h\), choose optimal plans between \(\mu_t\) and \(\mu_{t+h}\) and view \((y-x)/h\) as a discrete velocity. The \(L^p\) norm of this velocity is \(W_p(\mu_t,\mu_{t+h})/|h|\), which is locally bounded by the metric derivative. Weak compactness in \(L^p(\mu_t)\), together with differentiation of test-function integrals, produces a limiting field \(v_t\) satisfying the continuity equation. Lower semicontinuity gives \(\|v_t\|_{L^p(\mu_t)}\le|\dot\mu_t|\). Combined with the first inequality, equality follows for the minimal representative.
The superposition principle
Superposition principle. If \((\mu_t,v_t)\) solves the continuity equation and \(\int_0^T\int|v_t|\,\mathrm{d}\mu_t\,\mathrm{d}t<\infty\), then there exists a probability measure \(\boldsymbol\eta\) on \(AC([0,T];\mathbb R^d)\) such that \[ (e_t)_\#\boldsymbol\eta=\mu_t,\qquad \dot\gamma_t=v_t(\gamma_t) \] for \(\boldsymbol\eta\)-almost every path and almost every time.
This theorem is a measure-valued extension of the method of characteristics. Even when the vector field has no unique classical flow, the solution is still a probability distribution over integral curves.
Minimal velocities and tangent directions
Velocity fields are not unique. A field \(w_t\) satisfying \[ \nabla\cdot(w_t\mu_t)=0 \] may be added without changing the curve. For \(p=2\), the minimal velocity is the orthogonal projection onto \[ T_{\mu_t}\mathcal P_2 =\overline{\{\nabla\phi:\phi\in C_c^\infty(\mathbb R^d)\}}^{L^2(\mu_t)}. \] Divergence-free components are orthogonal to all gradients and contribute kinetic energy without moving the measure at first order.
Translation, dilation, and displacement flows
Translation. If \(\mu_t=(x\mapsto x+ta)_\#\mu_0\), then \(v_t\equiv a\), the continuity equation holds, and \(|\dot\mu_t|=|a|\).
Dilation. If \(\mu_t=(x\mapsto e^t x)_\#\mu_0\), then \(v_t(x)=x\). Hence \[ |\dot\mu_t|^2=\int|x|^2\,\mathrm{d}\mu_t(x) \] when this field is the minimal \(L^2\) representative.
A displacement geodesic. If \(\mu_t=((1-t)\operatorname{id}+tT)_\#\mu_0\), then along the particle starting at \(x\), velocity is \(T(x)-x\). In Eulerian coordinates, \[ v_t=(T-\operatorname{id})\circ T_t^{-1} \] wherever \(T_t\) is invertible. Its kinetic norm is constant and equals \(W_2(\mu_0,\mu_1)\).