Let $\{X_n\}$ be a time-homogenuous markov chain, with state space $\mathbb{X}$ and transition matrix $P$. Let $C_1$ be a transient class and $C_2$ be a recurrent class and let also $x\in C_1,~y\in C_2$. Suppose that $C_1 \rightarrow C_2$, i.e. $p^{(n)}(x,y)>0$ for some postive integer $n$.
What is the methodology to determine the following probabilities?
(1) chain leaves (at some point) the state $x$ in favor of $y$, i.e. $P(X_n=y~|~X_0=x),$
(2) chain leaves the state $x$ in favor of class $C_2$, i.e. $P(X_n\in C_2~|~X_0=x)$.
Some thoughts. If $x$ belonged in $C_2$ then I would work on $C_2$, with the restriction $Q$ of the transition matrix on $C_2$. In that case $P(X_n=y)=\pi_n(y)$, where $\pi_n=\pi_0 Q^n$ ($\pi_0$ the initial distribution). But $C_1$ is transient, so things are quite different. I even don't know if there is a solution for every $n$ or if we can find only the limit $\displaystyle \lim_{n \rightarrow +\infty}P(X_n=y~|~X_0=x).$
Thanks a lot in advance for the help!