According to Wikipedia, the Feynman-Kac formula allows one to map
$\frac{\partial u}{\partial t}(x,t) + \mu(x,t) \frac{\partial u}{\partial x}(x,t) + \tfrac{1}{2} \sigma^2(x,t) \frac{\partial^2 u}{\partial x^2}(x,t) = 0$
onto SDEs of the form
$dX = \mu(X,t)\,dt + \sigma(X,t)\,dW$.
However, according to e.g. eq. (38) in this paper, the relevant Fokker-Planck equation would rather correspond to
$\frac{\partial u}{\partial t}(x,t) + \frac{\partial}{\partial x}\left[\mu(x,t) u(x,t)\right] + \frac{1}{2} \frac{\partial^2 }{\partial x^2}\left[\sigma^2(x,t)u(x,t) \right] = 0$.
How can they both be correct?