Find an SDE given a probability density function p(x,t)

Question

Usually, we start with an SDE for $X(t)$ and then try to solve it and find the corresponding probability density function $p(x,t)$. But what if we start with a pdf - can we find a corresponding SDE? How can this be done? Would we use the Fokker-Plank equation and try to "guess" the mean and diffusion coefficient? Or is there an approach that doesn't require guessing? Would such SDE be unique?