Stationary distribution of DTMC with infinite state space

Question

I am solving the stationary distirbution of a Discrete time Markov Chain with infinite state space. The state space is $\{\pi^H_0,\pi^H_1,...,\pi^L_0,\pi^L_1,...\}$. The transition matrix has structure. I eventually reduce the system linear equations into the following two series. But feel no clue about how to start to solve them.... \begin{align*} &\begin{cases} &\pi^H_0=a\pi^H_1\\ &\pi^H_1=a\pi^H_0+a\pi^H_2+c\pi^L_0\\ &\pi^H_n=b\pi^H_{n-1}+a\pi^H_{n+1},~\forall~n\geq 2 \end{cases}~~ \begin{cases} &\pi^L_0=c\pi^L_1\\ &\pi^L_1=b\pi^H_0+d\pi^L_0+c\pi^L_2\\ &\pi^L_n=d\pi^L_{n-1}+c\pi^L_{n+1},~\forall~n\geq 2 \end{cases}\\ &\sum_{n=0}^{\infty}(\pi^H_n+\pi^L_n)=1\\ &a+b=1,~c+d=1,~a,b,c,d~\mbox{are all positive constant }\{\pi^H_n\}_{n=0}^\infty~\mbox{and }\{\pi^L_n\}_{n=0}^\infty~\mbox{are all non-negative variables} \end{align*}

Answer 1

The recurrences in the third lines aren't coupled, so they can be solved with the standard ansatz $\pi_k^H=\lambda^k$:

$$ \lambda^n=b\lambda^{n-1}+a\lambda^{n+1}\;, $$

which yields the characteristic equation

$$ a\lambda^2-\lambda+b=0 $$

(and likewise for $\pi^L$). This yields two linearly independent solutions each, for a total of $4$ free coefficients. The first two lines allow you to express the values for $n=1,2$ in terms of the initial values for $n=0$, yielding one condition each. That leaves two degrees of freedom, and normalization reduces this to one. This might be eliminated if one of the characteristic values has magnitude greater than $1$ and thus isn't suitable for representing probabilities.