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

TL;DR

This paper investigates the regularity and fractal properties of two-sided whole-plane SLE curves parametrized by Minkowski content, establishing their optimal Hölder continuity and dimension scaling behavior.

Contribution

It proves the optimal Hölder continuity exponent for SLE curves parametrized by Minkowski content and characterizes the Hausdorff dimension of their images of deterministic sets.

Findings

SLE curves are locally α-Hölder continuous for any α<1/d.

For κ in (0,4], SLE curves are not locally 1/d-Hölder continuous.

Hausdorff dimension of the image of A under γ equals d times the Hausdorff dimension of A.

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 $\frac 1d$ 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 $A$.