This question was already answered here mathoverflow.net/questions/221585/…

Is the function $a$ dependent on $u$ or on $x$ only. In the former case it is not obvious to link it to stochastic differential equations easily as the PDE is not linear and your process $X_t$ will depend on $u$ which is unknown here. If $a$ depends only on $x$ you can represent it via Feyman-Kac and all what you need for the existence of the solution to the SDE equation on $X_t$ is that $a$ is Lipschitzian.

I was correcting the formula in the same time you posted your comment :)

You're right, I see that the purpose of the first part of imateapot answer is to justify that this conditional expectation is $g(X)$ measurable when $X$ and $g(X)$ are independent. I'll leave the answer like that so the mistake can be clear to the reader