Let's say there is thing with a probability distribution function $f(x,t)$ describing the probability amplitude of this thing as a function of both $x$, and of $t$. Also this same thing has a probability distribution function $g(x'(t),t)$ describing the probability amplitude of this thing as a function of both $x'(t)$, and $t$ with $x'(t)$ being the first derivative of $x$ with respect to $t$. Also this same thing has a probability distribution function $f(x,0)$ for the probability of this thing as a function of $x$, when $t=0$.
Given $g(x'(t),t)$, and $f(x,0)$ how do I find $f(x,t)$?