Expected occupation count before absorption

The expected number $u(i)$ of visits to a state $k$ before absorption satisfies the reward equationwith zero boundary data at absorbing states where counting stops.

For simple random walk on a finite connected loopless unweighted graph, start at $a\ne z$ and stop on first hitting $z$. Ifthen $v_{az}(z)=0$ and $L{v}_{az}={\delta }_a-{\delta }_z$ for the Graph Laplacian $L$. The incoming-visit balance equation is $g_{az}(u)=1_{\{u=a\}}+{\sum }_{w\sim u}{g}_{az}(w)/\mathrm{deg}(w)$ for $u\ne z$; at $z$ the Laplacian value is $-1$ because all its coordinates sum to zero. Thus $v_{az}$ is the voltage for unit current from $a$ to $z$, and $v_{az}(a)=R_{\mathrm{eff}}(a,z)$. In particular, the expected number of directed transitions $u\to w$ before $T_z$ is $v_{az}(u)$.

For the chain with $0$ absorbing, equal-probability nearest-neighbour moves between $1,2,3$, and a probability-$1/2$ self-loop at $3$, starting from $1$ the expected absorption time is $6$, the probability of visiting $3$ before $0$ is $1/3$, and the expected number of visits to $3$ before absorption is $2$.