Suppose we have some very imprecise knowledge of classical particle's coordinates and momentum: what we can only tell is the probability density to find it in some point of phase space. This is (almost?) all what is usually known by quantum state function.
For quantum particle, there's an equation, which governs such initial state — it's Schrödinger equation.
Is there any known equation, which would similarly govern evolution of classical particle in some external potential, given initial probability density in phase space?