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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.