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Time reversal

For a CTMC X, the time-reversed process is defined to be X ^ t = X T − t {\displaystyle {\hat {X}}_{t}=X_{T-t}}. By Kelly's lemma this process has the same stationary distribution as the forward process.

A chain is said to be reversible if the reversed process is the same as the forward process. Kolmogorov's criterion states that the necessary and sufficient condition for a process to be reversible is that the product of transition rates around a closed loop must be the same in both directions.