For a standard Brownian motion $W\_t$ and a given crossing probability $\alpha < 1$, I want to have a boundary function $f(t) > 0$, such that the probability that $W\_t$ ever crosses the boundary is bounded by the given probability, i.e.
$$ P(|W\_t| > f(t), \exists t > 0) \le \alpha $$
By Law of the iterated logarithm, any $f(t) = O(\sqrt{t\log\log t})$ does not meet this condition because
$$ \lim\sup\_{n\rightarrow \infty} \frac{W\_t}{\sqrt{t\log\log t}} = \sqrt{2} $$ if I get it right. 
For my case, I would like an $f(t) = o(t)$. In other words, I want to have a function such that 
$$ \lim\_{t\rightarrow \infty}\frac{f(t)}{t} = 0 $$.