Probability of a Wiener process staying within a wedge

Question

Given $m>0$, what is the probability that a one-dimensional Wiener process $W$ will satisfy

$$\left|W_t\right|\leq mt$$

at all times during the period $t \in [0,1]$?

Answer 1

The probability is zero, since the local modulus of continuity of Brownian motion when $t\to0$ is $\sim\sqrt{2t\log\log1/t}$, which is $\gg t$.