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

For standard Brownian motion started at zero, $H_a=\inf \{t\ge 0:B_t\ge a\}$ is finite almost surely for $a\ge 0$ andThe Brownian reflection principle proves finiteness. For $u\ge 0$, the Exponential martingale for Brownian motion stopped at $H_a\wedge t$ is bounded by $e^{ua}$. The dominated convergence theorem gives $\mathbb{E}{e}^{ua-u^2{H}_a/2}=1$, and taking $u=\sqrt{2\lambda }$ gives the transform.

For standard Brownian motion and $a>0$, $T_a=\inf \{t\ge 0:B_t\ge a\}$ is finite almost surely andThe Brownian reflection principle gives this distribution function. The Brownian scaling identity $T_{ca}\stackrel{d}{=}{c}^2{T}_a$ and the Strong Markov property show its probability distribution is a strictly stable distribution with index $1/2$.

Let $\tau$ be the first attainment of the maximum of standard Brownian motion on $[0,T]$, with $T>0$. Time reversal shows $\tau <T$ with probability one. Then $Y_t\le 0$ for $0\le t\le T-\tau$, an initial interval of positive length. Standard Brownian motion has probability zero of this property by the Brownian reflection principle, so $Y$ is not Brownian. Moreover $\tau$ is not a stopping time for the natural Brownian filtration: for $0<t<T$, $\mathbb{P}(\tau \le t\mid {\mathcal{F}}_t)=2\Phi ((M_t-B_t)/\sqrt{T-t})-1$, which is strictly between zero and one on $\{B_t<0\}$. The Strong Markov property is therefore inapplicable at this random time.

For standard Brownian motion, the Brownian reflection principle bounds the probability that its maximum up to $a^n$ exceeds $(1+\epsilon )\sqrt{2a^n\log \log (a^n)}$ by $2(n\log a{)}^{-(1+\epsilon {)}^2}$. This is summable for every $a>1$ and $\epsilon >0$. The Borel-Cantelli first lemma controls these geometric times, monotonicity of the normalizing function controls the intervening times, and countable choices $a\downarrow 1$, $\epsilon \downarrow 0$ give the bound. Equality requires a separate lower-bound proof.

A halo is associated with the first upcrossing of the linear spherical-collapse threshold by the smoothed density contrast as its variance increases. With a sharp-k smoothing filter, the Brownian reflection principle makes the crossing probability twice the one-scale Gaussian upper tail. This handles trajectories that crossed earlier but finish below the barrier and leads to the Press-Schechter halo mass function.

For $a,b>0$,This follows from the Brownian reflection principle weighted by the Cameron-Martin theorem for a linear drift.

Let $M_t={\sup }_{0\le s\le t}{B}_s$ for a standard Brownian motion. For $m\ge 0$, the Brownian reflection principle givesOn $b<m$, the pair $(B_t,M_t)$ therefore has joint probability density