= Solution
Integrating the continuity equation over state space and applying the <divergence theorem> gives
$$
\frac d{dt}\int P\,dV
=-\int\nabla\cdot\mathbf J\,dV
=-\int_{\partial V}\mathbf J\cdot d\mathbf S.
$$
Assume the <probability density function> and its derivatives decay sufficiently rapidly at infinity, or impose a <zero-flux boundary condition>. The boundary integral then vanishes, so $\int P\,dV$ is constant. Since it is initially one, \b[the distribution remains normalized for all time.]
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