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

Research paper by Dapeng Zhan

Indexed on: 18 Dec '18Published on: 17 Dec '18Published in: Probability Theory and Related Fields

#### 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.