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

• 成果类型: Article
• 署名作者: Zhan, Dapeng
• 署名单位: Michigan State University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-018-0895-0
• 发表日期: 2019
• 页码: 447-466
• 关键词: natural parameterization

摘要

We make use of the fact that a two-sided whole-plane Schramm-Loewner evolution (SLE kappa) curve gamma for kappa is an element of (0, 8) from infinity to infinity through 0 may be parametrized by its d-dimensional Minkowski content, where d = 1 + kappa/8, and become a self-similar process of index 1/d with stationary increments. We prove that such gamma is locally alpha-Holder continuous for any alpha < 1/d. In the case kappa is an element of (0, 4], we show that gamma is not locally 1/d-Holder continuous. We also prove that, for any deterministic closed set A subset of R, the Hausdorff dimension of gamma(A) almost surely equals d times the Hausdorff dimension of A.