I have a Markov Chain of $N$ states. Such states represent the energy levels in a molecule.


The states' connectivity is as follows:


1. States $j\in\{0,\ldots,N\}$ transition to $k\in\{\max(j-M,0),...,\min(j+M,N)\}\setminus j$, with $N\gg M$. The resulting transition matrix $\mathbf{P}$ has known terms $P\_{jk}$, and it is a hollowed band matrix where detailed balance is followed.
2. States $l\in\{L,...,N\}$ ALSO transition outside the Markov Chain, with rates $\mu\_l>0$. Every state $l$ retains transitions to the $k$ states described in the previous point. For instance, state $L+1$ has both $j$ and $l$ transitions.


I am interested in recovering the stationary distribution $\tilde\pi\_j$ of such Markov Chain. 


If $\mu\_l=0$, the Markov Chain has stationary distribution $\pi\_j$ known from detailed balance and probability closure condition: 
$$
\pi\_kP\_{kj}=\pi\_{j}P\_{jk},\quad \sum\_j\pi\_j=1
$$
writing $\pi\_j=\pi\_0P\_{0j}/P\_{j0}$ and substituting into $\sum\_j\pi\_j=1$, one obtains $\pi\_0$ and all the remaining values.


Now, when $\mu\_l>0$, the stationary distribution $\tilde\pi\_j$ is different. To tackle the problem, I have assumed that the $l$ states go to an absorbing state $J=N+1$, thus the transition matrix gains an absorbing state (one row and one column) with:
$$
P\_{lJ}=\mu\_l,\quad P\_{(j\setminus l)J}=0,\quad P\_{Jl}=0,\quad P\_{J(j\setminus l)}=0,\quad P\_{JJ}=1
$$
But from now, to the best of my knowledge, I do not know how to proceed. I have the feeling that by introducing $J$, I am rendering the Markov Chain not ergodic anymore (as all states are unreachable from $J$). I have the feeling that $\tilde \pi\_j=\delta(j-J)$, having $\delta(x)$ as the Dirac Delta Function.


I have read various sources (Snell, Montroll, Othmer) on Markov Chains and queuing theory, but all examples were not as specific as the one I have described above.



Stationary distribution of Markov Chain with departure
As the comments state, the resulting Markov Chain must have equilibrium distribution
$$
\tilde\pi\_j = \delta(j-J)
$$
since any random walk through the Chain will eventually lead to states $j\ge L$, and then to $J$. This is justified by the [Perron-Frobenius Theorem](https://en.wikipedia.org/wiki/Perron%E2%80%93Frobenius_theorem#Finite_Markov_chainshttps://en.wikipedia.org/wiki/Perron%E2%80%93Frobenius_theorem#Finite_Markov_chains) applied to the transition matrix $\mathbf{P}$. 


Alternatively, the inevitable eventuality that a random process will hit $j$ may be given by transitive [subshift of finite type](https://en.wikipedia.org/wiki/Subshift_of_finite_type) applied to the finite oriented graph $G$ generated by $\mathbf{P}$, or the adjacency matrix $\mathbf{A}(\mathbf{P})$ such that:
$$
a\_{ij}=H(p\_{ij})
$$
where $H$ is the Heaviside function $H(x>0)=1$. Since $G$ is strongly connected (there is a sequence of edges from any one vertex to any other vertex), then subshift of finite type on $G$ is transitive. 


There may be other tools to describe the existence of an equilibrium distribution for such processes, such as the [Krylov-Bogolyubov Theorem](https://en.wikipedia.org/wiki/Krylov%E2%80%93Bogolyubov_theorem). 


