Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 214 / 3 / b

Put $A=\{a,b\}$ and let ${\tau }_x^{+}=\min \{t\ge 1:X_t=x\}$. The conductance-hitting identity for an electrical network iswhere the vertices of $D$ are wired together. It follows by taking the hitting probability of $D$ before returning to $x$ as a voltage and computing its total current out of $x$.

Split the walk into successive excursions from $x$. The first excursion that hits $A$ determines whether $a$ or $b$ is hit first. Its conditional probability of hitting $a$ is at most the probability that an arbitrary excursion hits $a$, divided by the probability that it hits $A$. Hence