# Can an absorbing CTMC be reversible?

Can a CTMC with an absorbing state be reversible? I guess not, as the product of rates through any loop cannot be equal when the loop involves the absorbing state (Kolmogorov criterion). Is my intuition correct? Is there any way of formalising this intuition?

The rates for going from an absorbing state $i$ to another state $j$, $q_{ji}$ with $i\neq j$, are zero. Then the Kolmogorov's criterion is verified if $i$ is included in the loop, since you have zeros at both sides (the loop is closed, so whenever $i$ is included, both $q_{ij}\geq0$ and $q_{ji}=0$ have to be present). So, in principle, the CTMC could be reversible if the Kolmogorov's criterion is verified for all finite sequences of states excluding the absorbing ones.