Translation of a Dirac mass

For $s,t\in [0,T]$, any coupling of ${\delta }_{a+sb}$ and ${\delta }_{a+tb}$ is concentrated at $((a+sb),(a+tb))$, hence [F1] gives $W_2({\mu }_s,{\mu }_t)=\mathrm{\mid }t-s\mathrm{\mid }\mathrm{\mid }b\mathrm{\mid }$. For every bounded continuous f, $f(a+tb)$ is continuous, so the curve is narrowly continuous. Thus it is $A{C}^2$ with constant metric derivative $\mathrm{\mid }b\mathrm{\mid }$ (the endpoint difference quotient is exactly $\mathrm{\mid }b\mathrm{\mid }$ at every interior time).

For $\phi \in C_c^{\mathrm{\infty }}((0,T)\times {\mathbb{R}}^d)$, the chain rule gives $\frac{d}{dt}\phi (t,a+tb)={\mathrm{\partial }}_t\phi (t,a+tb)+{\mathrm{\nabla }}_x\phi (t,a+tb)\cdot b\mathrm{.}$ The left side integrates to zero over $[0,T]$ because the test vanishes near both time endpoints; this is exactly the weak continuity-equation identity for ${\mu }_t={\delta }_{a+tb}$ and $v_t=b$. Its action is ${\int }_0^T\mathrm{\mid }b{\mathrm{\mid }}^{2}dt=T\mathrm{\mid }b{\mathrm{\mid }}^2$. By [F3] and step 1.1, this equals ${\int }_0^T\mathrm{\mid }{\dot{\mu }}_t{\mathrm{\mid }}^{2}dt$, the minimum action.