Harmonic Function Method for Splitting Probability and Hitting Time

A Markov chain on $\{0, 1, 2, 3, 4, 5\}$ has absorbing states $0$ and $5$. For interior states: $$p(1,0) = \tfrac{1}{4}, \quad p(1,2) = \tfrac{3}{4},$$ $$p(2,1) = \tfrac{1}{3}, \quad p(2,3) = \tfrac{2}{3},$$ $$p(3,2) = \tfrac{1}{2}, \quad p(3,4) = \tfrac{1}{2},$$ $$p(4,3) = \tfrac{2}{3}, \quad p(4,5) = \tfrac{1}{3}.$$

Let $T = \inf\{n \ge 0 : X_n \in \{0, 5\}\}$.

(a) A function $f$ on $\{0,\ldots,5\}$ is *harmonic* on $\{1,2,3,4\}$ if $f(i) = \sum_j p(i,j) f(j)$ for $i \in \{1,2,3,4\}$. Find the harmonic function with $f(0) = 0$, $f(5) = 1$, and use it to compute $P(X_T = 5 \mid X_0 = 2)$.

(b) Compute $E[T \mid X_0 = 2]$.