I don't quite understand a property of the Wiener process. Such process has the property that
$W(t) - W(s) \sim \mathcal{N}(0, t-s)$
where $t > s > 0$. What I don't understand is this. As the distance between $t$ and $s$ increases you have a gaussian variable with a variance that increases with it, so you would expect "big variations". However, if $t-s$ is large enough, w.h.p. somewhere between $t$ and $s$, say at t', the process crosses the zero, i.e., W(t')=0.
So my question is this: if $s=0$ you can say that at time $t>>1$, the process will be likely to be very far from zero, because the variance is $t$. However, there is a t' >>1, with t'