On Brownian motion, simple paths, and loops

• 成果类型: Article
• 署名作者: Sapozhnikov, Artem; Shiraishi, Daisuke
• 署名单位: Leipzig University; Kyoto University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-017-0817-6
• 发表日期: 2018
• 页码: 615-662
• 关键词: erased random-walks; exponent

摘要

We provide a decomposition of the trace of the Brownian motion into a simple path and an independent Brownian soup of loops that intersect the simple path. More precisely, we prove that any subsequential scaling limit of the loop erased random walk is a simple path (a new result in three dimensions), which can be taken as the simple path of the decomposition. In three dimensions, we also prove that the Hausdorff dimension of any such subsequential scaling limit lies. We conjecture that our decomposition characterizes uniquely the law of the simple path. If so, our results would give a new strategy to the existence of the scaling limit of the loop erased random walk and its rotational invariance.