On the Escape of a Random Walk From Two Pieces of a Tripartite Set

Let $\{A,B,C\}$ be a partition of a sample space $\mathrm{\Omega }$. For a random walk $S_n=x+{\sum }_{j=1}^n{X}_j$ starting at $x\in A$, we find estimates for the Green’s function $G_{A\cup B}(x,y)$ and the hitting time $E^x(T_C)$ for $x,y\in A\cup B$, with interest in the case where $C$ “separates” $A$ and $B$ in a sense (e.g. the probability of jumping from $A$ to $B$, or vice versa, before hitting $C$, is small).