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 $\pi_{1} = \pi_{2}$.
