Optimal Hölder continuity and dimension properties for SLE with Minkowski content parametrization

Abstract

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 $\frac{1}{d}$ with stationary increments. We prove that such $\gamma$ is locally $\alpha$-Hölder continuous for any $\alpha <\frac{1}{d}$. In the case $\kappa \in (0,4]$, we show that $\gamma$ is not locally $\frac{1}{d}$-Hölder 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.