A bit of background for the question. I am self-studying some Markov Chain theory, and got stuck in a particular question. So here are the definitions I am working with.
We have a Markov chain $M$ on some countable state space $V$ (say vertices of a graph), with transition probabilities properly defined. Then the probability that the hitting time of $y$ from $x$ ($x,y\in V$) is finite is written as $$\rho_{xy}:=\mathbb P_x(\tau_y^+<\infty)$$
I am trying to prove the equivalence $$\rho_{xx}<1\iff \rho_{xy}<1$$
I did the if part (hopefully I am correct). This is how I proceeded
\begin{align*}\rho_{xx}&=\mathbb P_x(\tau_x^+<\infty\mid \tau_y^+<\infty)\mathbb P_x(\tau_y^+<\infty)+\mathbb P_x(\tau_x^+<\infty\mid \tau_y^+=\infty)\mathbb P_x(\tau_y^+=\infty)\\ &\le \mathbb P_x(\tau_x^+<\infty\mid \tau_y^+<\infty)\mathbb P_x(\tau_y^+<\infty)+\mathbb P_x(\tau_y^+=\infty)\\&=\mathbb P_x(\tau_x^+<\infty\mid \tau_y^+<\infty)\rho_{xy}+(1-\rho_{xy}) \end{align*} Then by the magic of Markov chains,
$$\rho_{xx}\le \rho_{yx}\rho_{xy}+(1-\rho_{xy})=1-\rho_{xy}(1-\rho_{yx})$$ and hence $\rho_{xx}<1$.
But I am hving a bit of trouble with the converse. I am trying a similar method
\begin{align*} \rho_{xy}&=\mathbb P_x(\tau_y^+<\infty\mid \tau_x^+<\infty)\mathbb P_x(\tau_x^+<\infty)+\mathbb P_x(\tau_y^+<\infty\mid \tau_x^+=\infty)\mathbb P_x(\tau_x^+=\infty)\\&=\mathbb P_x(\tau_y^+<\infty)\rho_{xx}+\mathbb P_x(\tau_y^+<\infty\mid \tau_x^+=\infty)(1-\rho_{xx})\\&=\rho_{xy}\rho_{xx}+\mathbb P_x(\tau_y^+<\infty\mid \tau_x^+=\infty)(1-\rho_{xx}) \end{align*}
Now, here if I naively put $0\le\mathbb P_x(\tau_y^+<\infty\mid \tau_x^+=\infty)\le 1$, then I get $$\rho_{xy}\le \rho_{xy}\rho_{xx}+(1-\rho_{xx})\Rightarrow \rho_{xy}\le 1$$ a pesky equality sign which I wanted to disprove!
I am thinking of somehow showing $\mathbb P_x(\tau_y^+<\infty\mid \tau_x^+=\infty)<1$ (which solves the problem), but I am drawing up a blank.
If this is true, a proof would be great, and if not, is there any other way to prove the converse? Any help is greatly appreciated!