I'm writing for my final year project as an undergraduate. According to the "Fokker-Planck equation" wikipedia page, if you have a Ito process $X_t$ described by the SDE $$dX_t = \mu(X_t,t) \, dt + \sigma(X_t,t) \, dW_t,$$ with $W_t$ a Brownian motion, then the probability density function of $X_t$, $p(x,t)$, is given by the Fokker-Planck equation: $$\frac{\partial}{\partial t}p(x,t) = -\frac{\partial}{\partial x}[\mu(x,t)p(x,t)]+\frac{\partial^2}{\partial x^2}[D(x,t)p(x,t)],$$ where $D(X_t,t)=\sigma^2(X_t,t)/2$.
My question is - if you come across a Fokker-Planck equation "in the wild", can you then take the above in the other direction, and associate the solution of this with the probability density function of an appropriate Ito process? If so, any directions to a proper reference would be incredibly helpful, as I'm struggling to find any reading on the basics of this equation, other than on Wikipedia. Many thanks!