Continuity and estimates of the Liouville heat kernel with applications to spectral dimensions

• 成果类型: Article
• 署名作者: Andres, Sebastian; Kajino, Naotaka
• 署名单位: University of Bonn; Kobe University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-015-0670-4
• 发表日期: 2016
• 页码: 713-752
• 关键词: GAUSSIAN MULTIPLICATIVE CHAOS

摘要

The Liouville Brownian motion (LBM), recently introduced by Garban, Rhodes and Vargas and in a weaker form also by Berestycki, is a diffusion process evolving in a planar random geometry induced by the Liouville measure , formally written as , , for a (massive) Gaussian free field X. It is an -symmetric diffusion defined as the time change of the two-dimensional Brownian motion by the positive continuous additive functional with Revuz measure . In this paper we provide a detailed analysis of the heat kernel of the LBM. Specifically, we prove its joint continuity, a locally uniform sub-Gaussian upper bound of the form for for each , and an on-diagonal lower bound of the form for , with heavily dependent on x, for each for -almost every x. As applications, we deduce that the pointwise spectral dimension equals 2 -a.e. and that the global spectral dimension is also 2.