So I understand the derivation of the Kolmogorov Forward and Backward Equations, but I don't quite understand the intuition. Here is from Stokey, 2008:
"The backward equation involves time $t$ and the initial condition $x$, with the current state $y$ held fixed. A similar PDE, the Kolmogorov forward equation (KFE), involves $t$ and $y$, with the initial state $x$ fixed. The forward equation is useful for characterizing the limiting distribution, if one exists."
In what situations do the two arrise? For instance, if I know the current state and am interested in a probability distribution over possible initial states, I use the backward equation. If I know the current state and am interested in the probability distribution over the state in the future, I use the forward equation. Is this correct?
On a more technical note, how does one define boundary conditions for these PDE's? Are the boundary conditions a result of the derivation? Perhaps Stokey is not the best reference on this topic...