Expected number of steps to return

Question

I have a simple random walk on a finite graph, starting in a how do i find the expected number of steps to return to a?

I know that to find the expected number of steps to return to say state 4 , we make state 4 absorbent and solve for (I-Q)^(-1), but i know that the state that we start in must be transient so i cannot make the state absorbent. Can anyone give me any advice?

Answer 1

Take one step. Now make $a$ absorbent.

Answer 2

If your graph is finite and $a$ is transient then it is nonrecurrent. This also means that it might happen with positive probability that you never return to $a$, i.e. if $\tau_a := \inf \{ n : X_n = a\}$ with $X_0=a$, then $P(\tau_a = \infty)>0$ and so $\mathbb E[\tau_a] = \infty$.