= Solution
Condition on the first jump, putting $h_{W_0}=1$. The <hitting probabilities> obey
$$
(\lambda+\alpha)h_{C_i}=\lambda h_{C_{i+1}}+\alpha h_{W_i}\quad(i\ge0),\qquad(\mu+\beta)h_{W_i}=\mu h_{W_{i-1}}+\beta h_{C_i}\quad(i\ge1).
$$
The relevant characterization is that a <hitting probability is the minimal nonnegative harmonic extension> of its target-boundary value. Thus among nonnegative solutions of these equations with $h_{W_0}=1$, choose the componentwise least one. Other harmonic solutions need not be probabilities. This also identifies the solution without imposing an unjustified boundary condition at infinity.
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