= Hitting probabilities for a velocity-switching Markov ladder
{title2=$\vartheta=\mu(\lambda+\alpha)/[\lambda(\mu+\beta)]$}
Consider states $C_i,W_i$ for <integer> levels $i\geq0$ and strictly positive rates $\lambda,\mu,\alpha,\beta$. From $C_i$ the chain moves to $C_{i+1}$ at rate $\lambda$ and to $W_i$ at rate $\alpha$. From $W_i$, $i\geq1$, it moves to $W_{i-1}$ at rate $\mu$ or to $C_i$ at rate $\beta$. At the reflecting boundary $W_0$ there is only the switch to $C_0$; no negative level is allowed. All rates are bounded, so the chain is <nonexplosive continuous-time Markov chain>[nonexplosive].
Let $h$ be the <probability> of hitting $W_0$, with $h_{W_0}=1$. <first-step analysis>[First-step equations] are
$$
(\lambda+\alpha)h_{C_i}=\lambda h_{C_{i+1}}+\alpha h_{W_i},\qquad
(\mu+\beta)h_{W_i}=\mu h_{W_{i-1}}+\beta h_{C_i}\quad(i\geq1),
$$
with the first equation also valid at $i=0$. Put $\vartheta=\mu(\lambda+\alpha)/[\lambda(\mu+\beta)]$. If $\mu\alpha<\lambda\beta$, then $\vartheta<1$ and
$$
h_{W_i}=\vartheta^i,\qquad
h_{C_i}=b\vartheta^i,\qquad b=\frac{\alpha(\mu+\beta)}{\beta(\lambda+\alpha)}<1.
$$
Substitution verifies both equations and the boundary. Eliminating the $C$ variables gives a second-order <recurrence> with modes $1,\vartheta^i$. The displayed decaying solution is the minimal nonnegative harmonic solution and hence the <Hitting probability>.
When $\mu\alpha\geq\lambda\beta$, the growing mode is excluded by boundedness; at equality the second mode is linear and is likewise excluded. The only bounded solution with the boundary value is one, proving almost-sure hitting. Since the specified reflected chain is <irreducible Markov chain>[irreducible], these cases classify it as <recurrent Markov chain>[recurrent] or <transient Markov chain>[transient] respectively. Specifying the boundary rule is essential for that <recurrence> statement; making $W_0$ absorbing would be a different chain.
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