Inverse-square Gaussian law of Brownian first passage
= Inverse-square Gaussian law of Brownian first passage
{title2=$\tau_x\overset d=x^2/N^2$}
The <Brownian reflection principle> gives $\mathbb P(\tau_x\leq t)=\mathbb P(|N|\geq x/\sqrt t)$ for a standard normal random variable $N$. This is the distribution function of $x^2/N^2$. An independent <Brownian motion> observed at this hitting time has the <Cauchy distribution> with scale $x$.