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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.

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I'm not sure I understand what you are asking but here is a suggestion. If you have a probability distribution at $t=0$ for an n-dimensional random variable $\vec{a}$: $p(0,\vec{a})$ and you have a deterministic rule for the evolution of the state:

$$\vec{a}(\Delta t) = f_{\Delta t} (\vec{a}(0))$$

Then, the probability distribution for time $\Delta t$ is:

$$p(\Delta t,\vec{a}) = p(0,f^{-1}_{\Delta t}(\vec{a}))$$

That is, the value of the probability density function remains the same but at a different point in the state space.

The question gets a lot more interesting if you assume that the evolution of the state is non-deterministic and in that case you have a fully stochastic system.

In some fields of engineering (robotics, for example) this kind of system is very common. In estimation problems where you have an initial probability and assume a stochastic evolution of the state, and noisy measurements of the state, the problem is: how to estimate the initial and subsequent states based on the measurents? A classic tool for dealing with that problem assuming linearity and normal distributions is the Kalman filter .

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