Intuition of the Kolmogorov Equations 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...
I will try to answer to your last question. I did not read the paper but in models with higher dimensions, it is always difficult to find an analytical solution. 
If there exists an analytical solution (for a very basic model with a one-state variable), it is possible to derive the initial conditions for your control and state variable from your differential equations.

However, in systems where there does not exist an analytical solution, you are supposed to solve it numerically, in which case you must give a numerical value for one of your variables (or more until you find steady state values for all your variables.)

After, you can find the steady-state values of your variables in which case, your initial conditions should not be so far from the steady state level (otherwise, there would be some convergence problems in your model if you don't choose the appropriate initial values, close to steady state. It is another issue.)