Time reversal of a continuous-time Markov chain ($q_{ij}^{\ast }={\pi }_j{q}_{ji}/{\pi }_i$)

For an equilibrium chain with strictly positive stationary probabilities, reversed jump rates are $q_{ij}^{\ast }={\pi }_j{q}_{ji}/{\pi }_i$. Under detailed balance they equal the original rates, making the stationary path law reversible. Holding times and jump directions reverse together; this is more than symmetry of one-time distributions.