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 $A$.

Revisions were made according to the referees' comments

References in corpus (2)

• Convergence of radial loop-erased random walk in the natural parametrization
• Decomposition of Schramm-Loewner evolution along its curve