SLE loop measures

• 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/s00440-020-01011-7
• 发表日期: 2021
• 页码: 345-406
• 关键词: schramm-loewner evolution; minkowski content; reversibility; intersection; dimension; CURVES

摘要

We use Minkowski content (i.e., natural parametrization) of SLE to construct several types of SLE kappa loop measures for kappa is an element of(0,8). First, we construct rooted SLE kappa loop measures in the Riemann sphere (C) over cap, which satisfy Mobius covariance, conformal Markov property, reversibility, and space-time homogeneity, when the loop is parametrized by its (1+kappa-8) -dimensional Minkowski content. Second, by integrating rooted SLE kappa loop measures, we construct the unrooted SLE kappa loop measure in (C) over cap which satisfies Mobius invariance and reversibility. Third, we use Brownian loop measures to extend the rooted and unrooted SLE. loop measures from (C) over cap to subdomains of (C) over cap, which respectively satisfy conformal covariance and conformal invariance, and then further use the conformal invariance to extend unrooted SLE. loop measures to some Riemann surfaces. Finally, using a similar approach, we construct SLE. bubble measures in simply/multiply connected domains rooted at a boundary point. The space-time homogeneity of rooted SLE. loop measures in (C) over cap confirms a conjecture by Greg Lawler on the existence of such measures.