Can we say a Markov Chain with only isolated states is time reversible? By "isolated", I mean that each state of this Markov Chain has 0 probability to move to another state, i.e. transition probability $p_{ij} = 0$ for $ i \ne j$.  Thus, there isn't a unique stationary distribution.
But by definition, since for any stationary distribution $\pi$, we have
$$
\pi_{i}p_{ij} = 0 = \pi_{j}p_{ji}
$$
seems that we can still call this Markov Chain time reversible.
Is the concept "time reversible" still make sense in this situation?

A bit background, I was asked to find a Markov Chain, with certain restrictions, that is NOT time reversible.  But I found if the stationary distribution exist, the chain is always reversible.  So I guess that my be chance is that a chain who doesn't have unique stationary distribution.  Maybe in this situation we can't call the chain reversible.
 A: This Markov chain is reversible --- you've shown it yourself by showing that the detailed balance equation holds.
You can show that the detailed balance condition is equivalent to:


*

*$p_{ij}>0\Rightarrow p_{ji}>0$

*For every cycle $i_0,i_1,\dots,i_n,i_0$ in the set of states, the probability of 'going' in the cycle $i_0\rightarrow i_0$ is equal for clockwise and anticlockwise orientations.


Thus, you could say that your Markov chain is vacuously reversible as there are no cycles.
A: I do not think that constructing a markov chain with isolated states will give you a time irreversible markov chain. 
Consider the case when you have one isolated state. Since, an isolated state can never be reached from any other state, your chain is actually a union of two different markov chains.


*

*A markov chain that always stays in the isolated state (which is time reversible by definition) and

*A markov chain on the non-isolated states which may or may not be time reversible.
Thus, I do not think the above strategy will work. 
