= Solution
Here $\vartheta<1$ is equivalent to $\mu\alpha<\lambda\beta$, which gives $b<1$ from its explicit formula. Any nonnegative solution is of the form
$$
h^{(i)}=a\begin{pmatrix}1\\1\end{pmatrix}+(1-a)\vartheta^i\begin{pmatrix}b\\1\end{pmatrix},
$$
since its second component at level zero must be one. Letting $i\to\infty$ forces $a\ge0$ for nonnegativity. The choice $a=0$ is nonnegative, and subtracting it from any other such solution gives $a[(1,1)^T-\vartheta^i(b,1)^T]\ge0$. Therefore minimality selects
$$
\boxed{h_{C_i}=b\vartheta^i,\qquad h_{W_i}=\vartheta^i.}
$$
In particular the return probability after leaving $W_0$ is $b<1$, so \b[the irreducible chain is transient]. These formulas are the <hitting probabilities for a velocity-switching Markov ladder>.
Back to article page
