A transient chain can return with positive probability

Given: Assume AC. Let $p>q>0$ with $p+q=1$, and use the biased nearest- neighbor kernel on $E=\mathbb{Z}$,

For each fixed $x\in \mathbb{Z}$, let ${\mathbb{P}}_x$ be the canonical law with $X_0=x$, put $r=q\mathrm{/}p$, and let $T_x^{+}=\inf \{n\ge 1:X_n=x\}$.

AC is the principle that every family of nonempty sets has a choice function. (The Axiom of Choice)

For this biased kernel and each deterministic start $x$, the canonical chain law ${\mathbb{P}}_x$ exists under the stated AC assumption. (Green kernel of a biased integer walk)

For the same walk, the Green kernel satisfies $G(x,y)=\{\begin{array}{ll}{\displaystyle \frac{1}{p-q}},& y\ge x,\\ {\displaystyle \frac{r^{x-y}}{p-q}},& y<x\mathrm{.}\end{array}$ (Green kernel of a biased integer walk)

Under AC, recurrence is equivalent to divergence of the diagonal transition series. (Equivalent criteria for recurrence and transience)

A state is transient exactly when its positive-time return probability is strictly less than one. (Recurrent and transient states)

For a transient state with return probability ${\rho }_x={\mathbb{P}}_x(T_x^{+}<\mathrm{\infty })$, the expected visit count equals the diagonal transition series and is $1\mathrm{/}(1-{\rho }_x)$. (Equivalent criteria for recurrence and transience)