I would like to ask a question about Brownian motion:


Let $B$ be a standard brownian motion. How to show that $\mathbb P( \max\limits\_{0 \leq s \leq t} B(s) \in (a,b) )$ decreases exponentially in t with a, b fixed?



A question about brownian motions
The precise answer depends on the initial condition, let's assume you start the Brownian motion (with diffusion constant $D$) at time $t=0$ in some point $x\_0\in(a,b)$, then you ask for the probability
$$P(t)=\mathbb P[ \max\limits\_{0 \leq s \leq t} B(s) \in (a,b) ]$$ 
that the particle has not crossed the point $x=b$ for all times up to time $t$. The point $x=b$ functions as an absorbing boundary for the Brownian motion. The [survival probability](https://en.wikipedia.org/wiki/First-hitting-time_model) is given by
$$P(t)={\rm Erf}\,\left(\frac{b-x\_0}{2\sqrt{Dt}}\right)$$
For large $t$ this decays as as $1/\sqrt t$,
$$P(t)=\frac{b-x\_0}{\sqrt{\pi Dt}}+{\rm order}(1/t)$$
so much slower than exponentially.


