I am interested in the following topic: I would like to transform a physical motion X(t) in to a probability distribution P(X). Specifically, if I know the position of a particle at all times I want to know the probability that I will find it in a specific location (integrate P(x) over a small volume to get probability ). I am thinking of macroscopic systems for which we can write down a Hamiltonian. Of course, in my fantasy, the probability transform will take the form of an operator which is applied to the dynamic equation for X(t) to systematically produce a static diff eq for P(x). I have tried to define various limiting processes that would intuitively construct P(x), but none of them have led to results which I can inteperet as probability. It seems to me that this sort of analysis has obvious applications in Chaos theory, and potential applications towards moving beyond Quantum Mechanics. My question is: Where has this already been done? If it hasn't been done, I presume it can't be done.