= Solution
A <birth-death chain> on $\mathbb N_0$ has generator entries $q_{n,n+1}=\lambda_n$, $q_{n,n-1}=\mu_n$ for $n\geq1$, $\mu_0=0$, $q_{nn}=-(\lambda_n+\mu_n)$, and zero elsewhere. Assume the usual conservative nonexplosive chain. The <detailed balance for a birth-death process> are $\pi_n\lambda_n=\pi_{n+1}\mu_{n+1}$.
If they hold, each incoming term cancels an outgoing term, giving $\pi Q=0$ and thus invariance. Conversely the invariance equation at zero gives $\pi_0\lambda_0=\pi_1\mu_1$. At state $n$, write the stationarity equation as equality of the neighbouring probability currents
$$
\pi_{n-1}\lambda_{n-1}-\pi_n\mu_n
=\pi_n\lambda_n-\pi_{n+1}\mu_{n+1}.
$$
Induction from the zero boundary current gives <detailed balance for a birth-death process> for every edge. Thus \b[invariant measures are exactly the detailed-balance measures] in this standard birth–death setting. The argument is the generator characterization; a stationary probability law additionally needs finite normalization. For potentially explosive processes, formal generator balance alone would require additional boundary qualifications.
Back to article page
