Let $W_t$ be a Wiener Process. Then the brownian motion $B_t$ solves the SDE $$dB_t = dW_t$$ while the density $f(x,t)$ of $B_t$ with $B_0 = 0$ solves the diffusion equation $$\frac{\partial f}{\partial t} = \frac{1}{2} \frac{\partial^2f}{\partial x^2}.$$
Let $dX_t = a(X_t,t)dt + b(X_t,t)dW_t$ be another SDE. Are there positive results about the possibility of deriving a PDE from $a$ and $b$ which will be solved by the density of the solutions of the SDE (if existant)? At least if $a(X_t,t) = a(X_t)$ and/or $b(X_t,t) = b(X_t)$?