= Solution
Write $z=(x,v)$ and $B(z)=(v,-\nabla V(x))$. The <characteristic flow map> $\Phi_t$ solves $\dot z=B(z)$. Its vector field is globally <Lipschitz continuous>, so <Picard-Lindelöf theorem> gives a unique global flow with inverse $\Phi_{-t}$. Also $\operatorname{div}_z B=0$.
The proposed <Dirac measure> is the unit mass at $\Phi_t(z_0)$. For every compactly supported smooth test function $\psi$,
$$
\frac d{dt}\langle f_t,\psi\rangle=\frac d{dt}\psi(\Phi_t(z_0))
=B(\Phi_t(z_0))\cdot\nabla\psi(\Phi_t(z_0))=\langle f_t,B\cdot\nabla\psi\rangle.
$$
This is the <weak formulation> of $f_t+\operatorname{div}(Bf)=0$, equal to the <Liouville equation> because $\operatorname{div}B=0$. The initial unit mass is recovered at $t=0$. The characteristic equations are precisely $\dot{\widehat x}=\widehat v$, $\dot{\widehat v}=-\nabla V(\widehat x)$; the time-derivative dots, lost in the converted text, are present in the PDF.
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