I have a questions: We said that if we have a positive recurrent Markov chain, then there is a unique stationary distribution.
1.) Does this mean that if I have several positive recurrent classes, then each one corresponds to a separate stationary distribution? ( This is what my intuition tells me)
2.) Furthermore: If we have two disjoint sets $A \cap B = \emptyset$, $q(x) = \mathbb{P}_x(\tau_A > \tau_B)$ then we have $q|_A(x)=0$, $q|_B(x)=1$ and $(Id-P)q(x)=0$, where $P$ is the transition matrix. ($\tau$ is the recurrence time)
Now I was wondering: Does the converse also hold, so if we have a $q$ that satisfies $q|_A(x)=0$, $q|_B(x)=1$ and $(Id-P)q(x)=0$, where $P$ is the transition matrix, can we infer from this that $q(x) = \mathbb{P}_x(\tau_A > \tau_B)$.
If anything is unclear, please let me know.
Re 2., note that the difference of the functions is harmonic on the complement of A and B and zero on A and B hence the maximum principle shows that it is zero everywhere, QED.