Proposition 1.1 — the time-reversed process keeps $\pi$ in equilibrium

Let $(X(t), t \in \mathbb{R})$ be a stationary Markov process on a finite state space $S$ with transition rates $q$ and a strictly positive equilibrium distribution $\pi$, so that $\pi$ satisfies the equilibrium equations for $q$. Proposition 1.1 of Kelly and Yudovina states that the reversed process $Y(t) = X(-t)$ is again a stationary Markov process, with the same equilibrium distribution $\pi$ and with transition rates

The two algebraic assertions that content amounts to, and which Exercise 1.4 asks the reader to check, are formalized here:

1. the reversed rates have the same total rate out of every state,

so a jump of the reversed process occurs at the same rate as a jump of the original; 2. $\pi$ satisfies the equilibrium equations for $q'$, so $\pi$ is again an equilibrium distribution for the reversed process.

Together these say that $q'$ is a legitimate rate matrix with the same holding-time parameters as $q$, and that reversing a stationary process does not disturb its equilibrium distribution.

Formalization Note The state space is taken finite so that every sum appearing is a finite sum; the positivity of $\pi$ is what makes the quotient defining $q'$ meaningful.