Brownian barrier survival asymptotic
= Brownian barrier survival asymptotic
{c}
{title2=$\mathbb P(\sup_{s\leq t}B_s\leq h)\sim h\sqrt{2/\pi}\,t^{-1/2}$}
For standard one-dimensional <Brownian motion> and a fixed $h>0$, the <Brownian reflection principle> gives the survival <probability> $2\Phi(h/\sqrt t)-1$. The <standard normal distribution function> has derivative $1/\sqrt{2\pi}$ at zero, giving the displayed asymptotic. Thus the persistence exponent is one half, with leading constant $h\sqrt{2/\pi}$.