Brownian barrier survival asymptotic ($\mathbb{P}({\sup }_{s\le t}{B}_s\le h)\sim h\sqrt{2/\pi }{t}^{-1/2}$)

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 $h\sqrt{2/\pi }$.