 Solution

Choose $0<\alpha <\epsilon /p$. At dyadic level $m$, Markov inequality and a union bound giveuniformly in $n$. Summing over $m$ shows that, outside a set of probability at most $C_0{C}^{-p}$, all dyadic increments obey this bound. Chaining dyadic approximations and using continuity givesfor all $s,t$. Since $X_0^n=0$, the paths then lie in the stated compact Hölder set by the Arzela-Ascoli theorem. Taking $C$ large proves tightness.