Brownian paths are locally Holder below one half

For each integer $m\ge 2$, [F1]--[F3] give a continuous modification $Y^{(m)}$ of $B$ and a probability-one event $H_m$ on which its paths are locally Hölder for every exponent below $(m-1)\mathrm{/}(2m)$. Use [F7] to select one such pair for each $m$.

Let $C$ be the probability-one continuity event for $B$ from [F1]. For each $m$ and nonnegative rational $q$, modification gives $P(B_q=Y_q^{(m)})=1$. By countability and subadditivity in [F4], the intersection $H=C\cap {\bigcap }_{m\ge 2}{H}_m\cap {\bigcap }_{m\ge 2}{\bigcap }_{q\in {\mathbb{Q}}_{\ge 0}}\{B_q=Y_q^{(m)}\}$ has probability one.

Fix $\omega \in H$ and $m\ge 2$. On every interval $[0,N]$, the two real functions $t\mapsto B_t(\omega )$ and $t\mapsto Y_t^{(m)}(\omega )$ are continuous and agree on the dense rational subset. By [F4]--[F5] they agree everywhere on $[0,N]$, hence on $[0,\mathrm{\infty })$. Therefore $B(\omega )$ inherits all local Hölder exponents below $(m-1)\mathrm{/}(2m)$.

Given $0<\gamma <1\mathrm{/}2$, [F6] supplies an integer $m\ge 2$ so large that $1\mathrm{/}(2m)<1\mathrm{/}2-\gamma$, equivalently $\gamma <(m-1)\mathrm{/}(2m)$. Step 3.1 then gives the displayed Hölder bound on every $[0,T]$, with its constant allowed to depend on $\omega ,T,\gamma$. The same event $H$ works for all uncountably many $\gamma$, because only the countable integer family was intersected. AC is used exactly at step 1.1 and through the normal-law content of [F1]--[F2].