Here's part of a [wiki page](https://en.wikipedia.org/wiki/Kolmogorov%27s_criterion#Continuous-time_Markov_chains) John mentioned in his comment he linked to above.

**Continuous-time Markov chains**

The theorem states that a continuous-time Markov chain with transition rate matrix Q is reversible if and only if its transition probabilities satisfy[1]

$$ q_{j_1 j_2} q_{j_2 j_3} \cdots q_{j_{n-1} j_n} q_{j_n j_1} = q_{j_1 j_n} q_{j_n j_{n-1}} \cdots q_{j_3 j_2} q_{j_2 j_1} $$

for all finite sequences of states

$$ j_1, j_2, \ldots, j_n \in S $$

References

* Kelly, Frank P. (1979). Reversibility and Stochastic Networks. Wiley, Chichester. pp. 21–25.

By the way, this book is online [here](http://www.statslab.cam.ac.uk/~frank/BOOKS/book/whole.pdf)

**Continuous-time Markov chains**

The theorem states that a continuous-time Markov chain with transition rate matrix Q is reversible if and only if its transition probabilities satisfy[1]

$$ q_{j_1 j_2} q_{j_2 j_3} \cdots q_{j_{n-1} j_n} q_{j_n j_1} = q_{j_1 j_n} q_{j_n j_{n-1}} \cdots q_{j_3 j_2} q_{j_2 j_1} $$

for all finite sequences of states

$$ j_1, j_2, \ldots, j_n \in S $$

References

* Kelly, Frank P. (1979). Reversibility and Stochastic Networks. Wiley, Chichester. pp. 21–25.

By the way, this book is online [here](http://www.statslab.cam.ac.uk/~frank/BOOKS/book/whole.pdf)