= First-passage Laplace transform for Brownian motion with drift
{title2=$\mathbb E e^{-\lambda S}=e^{-y(a+\sqrt{a^2+2\lambda})}$}
For $Y_t=y+B_t+at$, $y>0$, and $S=\inf\{t\ge0:Y_t=0\}$,
$$
\mathbb E e^{-\lambda S}=\exp[-y(a+\sqrt{a^2+2\lambda})],\qquad\lambda>0,
$$
where $e^{-\lambda\infty}=0$. The <diffusion generator> is $\frac12\partial_{yy}+a\partial_y$. 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 $e^{-2y\max(a,0)}$.
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