= Solution
A <continuous-time Markov chain> started with distribution $\pi$ is <reversible> if its finite-dimensional distributions are unchanged by reversing time: for every $T>0$, $(X_t:0\leq t\leq T)$ and $(X_{T-t}:0\leq t\leq T)$ have the same law. In particular, reversibility includes <stationarity>. The <detailed balance> equations are
$$
\boxed{\pi_iq_{ij}=\pi_jq_{ji}\quad(i,j\in I).}
$$
For a finite <continuous-time Markov chain>, these equations are equivalent to reversibility with initial distribution $\pi$. Equivalently, they imply $\pi_iP_{ij}(t)=\pi_jP_{ji}(t)$ for every transition matrix $P(t)=e^{tQ}$.
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