I am currently reading a paper about planar Brownian Motion (BM) on a torus $\mathbb T_2$.
Setup: Let $x\in\mathbb T_2$ and $r_i:=R(\frac\epsilon R)^{\frac i K}$ for $i=0,1,\dots,K.$ Then $\partial B_{r_i}(x)$ are circles that are decreasing in size when $i$ is increasing. The smallest circle is $\partial B_\epsilon(x)$. Now take a Brownian Motion that starts somewhere in a neighborhood of $x$ and track the $i$ whenever BM hits $\partial B_{r_i}$ (but do not track consecutive hits of the same circle). Those numbers $i$ then give us our Simple Random Walk (SRW).
Question: I am trying to verify this statement in the paper:
First, it can be checked via Doob’s h-transform that the expected number of excursions from $l$ to $l + 1$ performed by a SRW started at 1 and stopped at 0 and conditioned not to hit $K$, is approximately $(1 − \frac{l + 1}K)^2$.
Unfortunately, I haven't really made much progress since I haven't even found good literature on Doob's h-transform. So if you could name one or two references on that, that'd be great. Or I'd really appreciate a proof (preferably with simpler methods).