Consider a random walk on some integer domain $[0, L]$, where the walk is initialized at some position $0 < k \leq L$. The $+1$ step probability is $p$, the $-1$ step probability is $(1-p)$, the position $L$ is a fully reflecting barrier, and the origin is a partially reflecting barrier with a probability $A$ for adsorption. To clarify what occurs at the boundaries, if the walker is at position $L$, it must take a $-1$ step with a probability of one, and if the walker is at the origin, it takes a $+1$ step with probability $(1-A)$ and is absorbed with probability $A$.
I'm hoping for a literature reference (or, if necessary, a calculation) for the mean duration of the above walk, and better yet, some idea of what the distribution looks like for the number of steps. Please note that what I'm looking for - a calculation for the mean adsorbence time, and an associated distribution for the number of steps provided one partially adsorbing boundary and one fully reflecting boundary, as described above - is presumably distinct from a calculation of the mean duration of a walk conditioned on eventual adsorption at one of the boundaries.
Here's a brief summary of the results in the literature that I am aware of:
[Parzen, 1962], pg. 233, eq. 6.32 gives a derivation for the probability of a walk, starting at some position $k$, adsorbing at $0$ before visiting $L$. Pg. 240, eq. 7.16 also provides a result for the mean time for adsorption at either $0$ or $L$, assuming the walk starts at some $0 < k < L$.
From [Percus, O.E., Adv. Appl. Prob. 17, pp. 594 - 606 (1985)] partially reflecting boundaries are introduced at $0$ and $L$ (here, with the same probability of adsorbing a walker), and exact probabilities of adsorbing at either boundary are provided, as well as calculations for the mean number of steps before adsorption.
[El-Shehawey, M.A., J. Phys. A: Math Gen. 33, pp. 9005 - 9013 (2000)] extends the work of Percus by considering the case where the adsorption probabilities at $0$ and $L$ are not equivalent. The probabilities in this work seem to agree with those of [Percus, 1985] up to a few typos. However, [Nandipati et. al., Phys. Rev. B 81, 235415 (2010)] notes an error with the conditional first passage times due to a mistreatment of boundary conditions, but only provides a correction for the case where $p = (1-p)$. It's not, however, clear to me where exactly this error occurs. Is anyone aware of a follow-up that addresses this matter?
In none of the above derivations can we treat the case where one boundary is fully reflecting.
(As a quick question, is it 'standard' to count a visit to a boundary without adsorbence as a step?)