Extremal geometry of a Brownian porous medium

• 成果类型: Article
• 署名作者: Goodman, Jesse; den Hollander, Frank
• 署名单位: Leiden University - Excl LUMC; Leiden University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-013-0525-9
• 发表日期: 2014
• 页码: 127-174
• 关键词: vacant set; motion; times

摘要

The path of a Brownian motion on a -dimensional torus run for time is a random compact subset of . We study the geometric properties of the complement as for . In particular, we show that the largest regions in have a linear scale , where is the capacity of the unit ball. More specifically, we identify the sets for which contains a translate of , and we count the number of disjoint such translates. Furthermore, we derive large deviation principles for the largest inradius of as and the -cover time of as . Our results, which generalise laws of large numbers proved by Dembo et al. (Electron J Probab 8(15):1-14, 2003), are based on a large deviation estimate for the shape of the component with largest capacity in , where is the Wiener sausage of radius , with chosen much smaller than but not too small. The idea behind this choice is that consists of lakes, whose linear size is of order , connected by narrow channels. We also derive large deviation principles for the principal Dirichlet eigenvalue and for the maximal volume of the components of as . Our results give a complete picture of the extremal geometry of and of the optimal strategy for to realise extreme events.