Brownian Hölder regularity

On every compact time interval, a Brownian path is almost surely a Hölder continuous function of every exponent $\alpha <1/2$, and of no exponent $\alpha \ge 1/2$. In particular, $B_t=o(t^{\gamma })$ as $t\downarrow 0$ for every $\gamma <1/2$ after choosing a Hölder exponent strictly between $\gamma$ and $1/2$.