I am looking for a reference for the following property of Markov Chains. Let $(X_n)_{n\geq0}$ be a (time homogenous) Markov Chain with finite state space $S=\{1,2,\ldots,n\}$, $n\in\mathbb N$, and transition matrix $P=(p_{ij})_{i,j\in S}$. Let $\psi:S\rightarrow S$ be an automorphism on $S$ (a permutation of the states). Assume the stationary distribution $\pi=(\pi_1,\ldots,\pi_n)$ exists and is unique.
Define the equivalence relation $i\equiv_\psi j$ iff there exists $m\in\mathbb N$, such that $i=\psi^m(j)$. This relation decomposes $S$ into disjoint subsets. We write $[i]=\{j\in S: j=\psi^{m_i}(i)\}$.
The interesting question is then, whether we can derive a new Markov Chain on the decomposition of $S$. I found the following condition in an old set of slides: Let $j,j'\in[j]$ arbitrary. Then $(X_n)_n$ is exactly lumpable, iff $$\sum_{i\in[i]}p_{ij}=\sum_{i\in[i]}p_{ij'}$$ for all choices of equivalence classes $[i]$ and $[j]$.
A consequence is that $\forall i,i'\in[i]:\pi_i=\pi_{i'}$. The existence of a unique stationary distribution is necessary condition for the existence of such a lumping.
The problem now lies in the definition of the new transitions. It is given as $$p_{[i][j]}=\frac{\#[j]}{\#[i]}\sum_{i\in[i]}p_{ij}.$$
This is somewaht troublesome, as the row sum does not need to be 1. That works for continuous time if we looked at the transition rates, only.
So my question is twofold:
- Is there a reference (other than Peter Buchholz) to exact lumpability
- Is there a reference to the discrete time case as above?
I have found, that concurrency theory seems to be touching upon similar topics... But I haven't studied it too deeply and would not know where to start.