= Hitting probabilities for a velocity-switching Markov ladder
{title2=$\vartheta=\mu(\lambda+\alpha)/[\lambda(\mu+\beta)]$}
On nonnegative levels, a chain moves right at rate $\lambda$ in state $C$, left at rate $\mu$ in state $W$, and switches $C\to W$ at rate $\alpha$ and $W\to C$ at rate $\beta$. The minimal nonnegative solution for hitting $W_0$ is identically one when $\mu\alpha\ge\lambda\beta$. When $\mu\alpha<\lambda\beta$, it is $h_{W_i}=\vartheta^i$ and $h_{C_i}=b\vartheta^i$, where $b=\alpha(\mu+\beta)/[\beta(\lambda+\alpha)]<1$. These cases give recurrence and transience respectively.
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