It is well-known that the Brownian motion (Wiener process) is almost sure locally $\alpha$-Holder for any $\alpha<1/2$. That is, with probability 1 $$ \sup_{t,s\in[0,1]}\frac{|W_t-W_s|}{|t-s|^{\alpha}}<\infty. $$

On the other hand, Brownian motion is clearly **not** globally $\alpha$-Holder. Indeed, it follows from the law of iterated logarithm, that
$$
\sup_{t\ge0}\frac{|W_t|}{|t|^{\alpha}}=\infty.
$$

One can play with the first inequality and with the help of Kolmogorov theorem and Borell-Cantelli obtain something that looks like ``global-local'' weighted Holder $$ \sup_{\substack{t,s\ge0\\ |t-s|\in[0,1]}}\frac{|W_t-W_s|}{(t^\delta\vee s^\delta\vee1)|t-s|^{\alpha}}<\infty. $$ for any arbitrarily small $\delta>0$.

My question is whether a global weighted Holder of the form $$ \sup_{t,s\ge0}\frac{|W_t-W_s|}{(t^\gamma\vee s^\gamma\vee 1)|t-s|^{\alpha}}<\infty $$ holds? What is the optimal weights exponent $\gamma$ (can one say any better than $\gamma=1$)? Does there exist a global version of Kolmogorov continuity theorem that one can directly apply?