does anyone have any suggestions regarding how to correctly treat the continuum limit of a random walk that has non-zero self-transition probabilities? To put this concretely, let's say that the walker has a forward-jumping probability of $\alpha$ and a backwards-jumping probability of $\beta$; I am interested in the case where
$$ \alpha + \beta < 1 $$
I am specifically interested in determining the first-passage time distribution for a random walker subject to these conditions, both on the full interval and on the half-interval subject to a reflection condition at $x=0$. This feels like a problem that has likely been solved somewhere before; however my best search efforts are coming up dry, and so I would appreciate any references or help. Thank you.