I'm having a lot of truble with those generators. So if we have an Itô diffusion (in $\mathbb R$ and $f,\sigma $ nice enough to have uniqueness of solution) $$dX_t=b(X_t)dt+\sigma (X_t)dB_t,$$ it's generator is given by $$Af(x)=b(x)f'(x)+\frac{1}{2}\sigma ^2(x)f''(x).$$
I know that will determine uniquely the distribution of $X_t$ but I really don't see how. Could someone explain how I can find the distribution from this ?
I tried to do a parallel with the ODE $dy=b(y)dx$. The generator would be $Af(x)=b(x)f'(x)$, but from this, I really don't see how to get information of $y$... something looks magic and not clear at all.