Solution
= Solution
The <telegraph process> has backward survival equations
$$
\boxed{\begin{aligned}
\partial_tQ^+&=s\partial_yQ^++\lambda(Q^--Q^+),\\
\partial_tQ^-&=-s\partial_yQ^-+\lambda(Q^+-Q^-).
\end{aligned}}
$$
Initially $Q^\pm(y,0)=1$. Only a left-moving trajectory reaches the target, while reflection reverses a right-moving velocity at the outer wall, so the hyperbolic boundary conditions are
$$
\boxed{Q^-(0,t)=0,
\qquad Q^+(L,t)=Q^-(L,t).}
$$