Volume growth and heat kernel estimates for the continuum random tree

• 成果类型: Article
• 署名作者: Croydon, David A.
• 署名单位: University of Warwick
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-007-0063-4
• 发表日期: 2008
• 页码: 207-238
• 关键词: parabolic harnack inequalities

摘要

In this article, we prove global and local (point-wise) volume and heat kernel bounds for the continuum random tree. We demonstrate that there are almost-surely logarithmic global fluctuations and log-logarithmic local fluctuations in the volume of balls of radius r about the leading order polynomial term as r similar to 0. We also show that the on-diagonal part of the heat kernel exhibits corresponding global and local fluctuations as t similar to 0 almost-surely. Finally, we prove that this quenched (almost-sure) behaviour contrasts with the local annealed (averaged over all realisations of the tree) volume and heat kernel behaviour, which is smooth.