Suppose $u(t,\omega), t \geq 0$ is a continuous time stochastic process with smooth paths so $\frac{d}{dt} u(t,\omega)$ exists for all $\omega$. Suppose you know the distribution of $\frac{d}{dt} u(t,\omega)$ and the initial distribution of $u$. Can you use this information to determine the distribution of $u(t,\omega)$ for all $t \geq 0$? That is, you know


1. $P(\frac{d}{dt}u(t,\omega) \leq A)$ for all $A \in R, t\geq 0$
2. $P(u(0,\omega) \leq A)$ for all $A \in R$.


Can you use 1 and 2 to determine $P(u(t,\omega) \leq B)$ for all $B \in R$? 


Can you go the other direction? That is, if you know $P(u(t,\omega) \leq A$ for all $A \in R$, what is $P(\frac{d}{dt} u(t,\omega) \leq B)$?



Distributions of Time Derivatives of Stochastic Processes
I would think the answer is "no" to both questions. 


Let me abbreviate $v(t)=du/dt$, so $u(t)=u(0)+\int\_0^t v(t')dt'$. Now ask for the expectation of $u^2(t)$. You'll need to know how $v(t')$ and $v(t'')$ are correlated for any $t',t''$ in the interval $(0,t)$. That information is not given.


Similarly, in the other direction, to find the distribution of the velocity $v(t)$ you need to know how the position $u(t)$ at nearby times is correlated, and that information is not given.


