= Brownian first-passage time
{c}
{title2=$T_a$}
For standard <Brownian motion> and $a>0$, $T_a=\inf\{t\geq0:B_t\geq a\}$ is finite <almost surely> and
$$
\mathbb P(T_a\leq t)=2\left(1-\Phi\left(\frac a{\sqrt t}\right)\right),\qquad t>0.
$$
The <Brownian reflection principle> gives this <distribution function>. The <Brownian scaling> identity $T_{ca}\overset d=c^2T_a$ and the <Strong Markov property> show its <probability distribution> is a <strictly stable distribution> with index $1/2$.
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