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Let there be two Markov processes on the same state space (which is countably infinite), but different transition matrices, denoted by $P_{1}$ and $P_{2}$. Assume positive recurrence, irreducibility and aperiodicity for both of them, so that a stationary distribution exists for both of them, denote them by $\pi_{1},\pi_{2}$ resp. Under what criteria do we have $\pi_{1} = \pi_{2}$. Obviously we should have $\pi_{1}P_{1} = \pi_{2}P_{2}$. What other stronger conditions are implied by $\pi_{1} = \pi_{2}$.

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Not sure if much is implied. Consider a finite state space (of size n), and consider the Markov Chain given by random walks on a (non-bipartite) d-regular graph on n vertices. Regardless of the structure of the graph (and the value d), the stationary distribution is uniform. This is a very large set of graphs, and there isn't much structural similarity between these graphs. –  Sesh Apr 14 '13 at 4:18
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Under the conditions you list, and with $\pi$ given, I think that $P$ can be described by specifying for each unordered pair $\lbrace i,j\rbrace$ a transition rate $P_{ij}$. The reverse rate $P_{ji}$ is determined by detailed balanced. But for each pair one rate is completely free and independent of all other rates so long as you satisfy ergodicity ($(P^n)_{ij}>0$ for some $n$). –  Yoav Kallus Apr 14 '13 at 4:51
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