= Brownian first-passage Laplace transform
{c}
{title2=$\mathbb E e^{-\lambda H_a}=e^{-a\sqrt{2\lambda}}$}
For standard <Brownian motion> started at zero, $H_a=\inf\{t\geq0:B_t\geq a\}$ is finite <almost surely> for $a\geq0$ and
$$
\mathbb E e^{-\lambda H_a}=e^{-a\sqrt{2\lambda}},\qquad\lambda\geq0.
$$
The <Brownian reflection principle> proves finiteness. For $u\geq0$, the <Exponential martingale for Brownian motion> stopped at $H_a\wedge t$ is bounded by $e^{ua}$. The <dominated convergence theorem> gives $\mathbb E e^{ua-u^2H_a/2}=1$, and taking $u=\sqrt{2\lambda}$ gives the transform.
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