First-passage Laplace transform for Brownian motion with drift ($\mathbb{E}{e}^{-\lambda S}=e^{-y(a+\sqrt{a^2+2\lambda })}$)

For $Y_t=y+B_t+at$, $y>0$, and $S=\inf \{t\ge 0:Y_t=0\}$,where $e^{-\lambda \infty }=0$. The diffusion generator is $\frac{1}{2}{\partial }_{yy}+a{\partial }_y$. Its bounded eigenfunction on the positive half-line with boundary value one is $u(y)=\exp [-y(a+\sqrt{a^2+2\lambda })]$, giving the formula through the discounted boundary-hitting representation. The zero-discount limit is the hitting probability $e^{-2y\max (a,0)}$.