= Scale increments for a birth-death chain
{title2=$w_{k+1}=(q_k/p_k)w_k$}
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 $x$ is
$$
\frac{\sum_{j=x+1}^{U}w_j}{\sum_{j=L+1}^{U}w_j}.
$$
This 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 $\rho=q_0q_1/(p_0p_1)$; geometric sums distinguish rightward, balanced and leftward escape.
// Target: markov-process.bigb
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