I came up with this question a while ago and haven't been able to gain any insight on it.
You are playing baseball. As a batter with finite vision capabilities, the only information you have about the ball when it leaves the pitcher's hand at $t = 0$ is some probability density function $\rho(t, \vec{x}, \vec{v})$ defined in 3-dimensional space. You assume a simple projectile motion path modeled by $\vec{x}'' = \vec{v}' = \vec{a}$, where $a$ is the acceleration due to gravity. How then, does $\rho$ evolve with time?
If the exact position and the velocity of the ball is known at some point in time, the solution is easy. However, in order to calculate the probability distribution for all time, you must consider an infinite number of paths the ball could take, depending on the ball's actual initial position and velocity, and then assign probability densities to each one. Unfortunately, this is where I am stuck.
I'm not looking for the answer to this specific question, but for the mathematical method required to solve for a time-evolving probability distribution function, where the equation of particle motion is known.