Scale increments for a birth-death chain ($w_{k+1}=(q_k/p_k)w_k$)

The interior equation for a harmonic function for a Markov chain on the integers gives $p_k(h(k+1)-h(k))=q_k(h(k)-h(k-1))$. Positive scale increments satisfy $w_{k+1}=(q_k/p_k)w_k$. For absorbing boundaries $L<U$, the lower exit probability from $x$ isThis satisfies the harmonic equation and the boundary values and therefore is the hitting probability. For parity-dependent probabilities, $w_{j+2}=\rho w_j$ with $\rho =q_0{q}_1/(p_0{p}_1)$; geometric sums distinguish rightward, balanced and leftward escape.

// Target: markov-process.bigb