1
$\begingroup$

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?

$\endgroup$
0
$\begingroup$

It's reversible with respect to the probability measure concentrated on the absorbing state.

There's a famous paper by Chung and Walsh on how to construct more sensible Markov chains that
may be considered the reversed process in this sort of situation.

$\endgroup$
0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.