Optimal H\"older Continuity and Dimension Properties for SLE with Minkowski Content Parametrization

We make use of the fact that a two-sided whole-plane Schramm-Loewner evolution (SLE$_\kappa$) curve $\gamma$ for $\kappa\in(0,8)$ from $\infty$ to $\infty$ through $0$ may be parametrized by its $d$-dimensional Minkowski content, where $d=1+\frac\kappa 8$, and become a self-similar process of index $\frac 1d$ with stationary increments. We prove that such $\gamma$ is locally $\alpha$-H\"older continuous for any $\alpha<\frac 1d$. In the case $\kappa\in(0,4]$, we show that $\gamma$ is not locally $\frac 1d$-H\"older continuous. We also prove that, for any deterministic closed set $A\subset \mathbb{R}$, the Hausdorff dimension of $\gamma(A)$ almost surely equals $d$ times the Hausdorff dimension of $A$.