Regularity of SLE in (t,κ) and refined GRR estimates

• 成果类型: Article
• 署名作者: Friz, Peter K.; Tran, Huy; Yuan, Yizheng
• 署名单位: Leibniz Association; Weierstrass Institute for Applied Analysis & Stochastics; Technical University of Berlin
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-021-01058-0
• 发表日期: 2021
• 页码: 71-112
• 关键词: continuity

摘要

Schramm-Loewner evolution (SLE kappa) is classically studied via Loewner evolution with half-plane capacity parametrization, driven by root kappa times Brownian motion. This yields a (half-plane) valued random field gamma = gamma (t, kappa;omega). (Holder) regularity of in gamma (center dot, kappa; omega), a.k.a. SLE trace, has been considered by many authors, starting with Rohde and Schramm (Ann Math (2) 161(2):883-924, 2005). Subsequently, Johansson Viklund et al. (Probab Theory Relat Fields 159(3-4):413-433, 2014) showed a.s. Holder continuity of this random field for kappa < 8(2 - root 3). In this paper, we improve their result to joint Holder continuity up to kappa < 8/3. Moreover, we show that the SLE kappa trace gamma(center dot, kappa) (as a continuous path) is stochastically continuous in. at all kappa not equal 8. Our proofs rely on a novel variation of the Garsia-Rodemich-Rumsey inequality, which is of independent interest.