= Solution
For the quadratic potential the <characteristic curves> are harmonic oscillations:
$$
\widehat x=x_0\cos t+v_0\sin t,\qquad \widehat v=v_0\cos t-x_0\sin t.
$$
The inverse is $x_0=x\cos t-v\sin t$, $v_0=v\cos t+x\sin t$. Therefore the <Liouville equation> has solution
$$
\boxed{f(x,v,t)=f_I(x\cos t-v\sin t,\ v\cos t+x\sin t).}
$$
This formula works for smooth data or by transport for locally integrable data, and for measures by pushforward. Although the quadratic $V$ itself is not globally Lipschitz as in the initial sufficient hypotheses, its gradient is globally Lipschitz, which is the property actually needed for this global flow.
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