I found a question regarding whether a process described by $dS(t)=\sigma dW(t)$ will breach a barrier of $H>S_0$, with $S_0$ being the initial level. This should amount to determining the probability that a Wiener process with $W_0=0$ will cross the x-axis by time $T$ as far I can tell.
By the reflection principal it seems $\mathbb{P}\left(\sup_{0<s<t}W(t)>a\right) = 2\mathbb{P}(W(t)>a)$, which in the case of the x-axis is 2*1/2=1.
Is this correct? There is a further question on the distribution of the time of breaching the barrier, so I find it difficult to believe this is correct.