The Brownian cactus II: upcrossings and local times of super-Brownian motion

• 成果类型: Article
• 署名作者: Le Gall, Jean-Francois
• 署名单位: Universite Paris Saclay
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-014-0569-5
• 发表日期: 2015
• 页码: 199-231
• 关键词: map; quadrangulations; trees; limit

摘要

We study properties of the random metric space called the Brownian map. For every , we consider the connected components of the complement of the open ball of radius centered at the root, and we let be the number of those connected components that intersect the complement of the ball of radius . We then prove that converges as to a constant times the density at of the profile of distances from the root. In terms of the Brownian cactus, this gives asymptotics for the number of points at height that have descendants at height . Our proofs are based on a similar approximation result for local times of super-Brownian motion by upcrossing numbers. Our arguments make a heavy use of the Brownian snake and its special Markov property.