LOGARITHMIC HEAT KERNEL ESTIMATES WITHOUT CURVATURE RESTRICTIONS

• 成果类型: Article
• 署名作者: Chen, Xin; LI, Xue-mei; Wu, Bo
• 署名单位: Shanghai Jiao Tong University; Imperial College London; Fudan University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/22-AOP1599
• 发表日期: 2023
• 页码: 442-477
• 关键词: quasi-invariance theorem; path space; sobolev inequalities; diffusion kernels; gradient estimate; brownian-motion; spectral gaps; derivatives; completeness; asymptotics

摘要

The main results of the article are short time estimates and asymptotic estimates for the first two order derivatives of the logarithmic heat kernel of a complete Riemannian manifold. We remove all curvature restrictions and also develop several techniques.A basic tool developed here is intrinsic stochastic variations with pre-scribed second order covariant differentials, allowing to obtain a path inte-gration representation for the second order derivatives of the heat semigroup Pt on a complete Riemannian manifold, again without any assumptions on the curvature. The novelty is the introduction of an 62 term in the variation allowing greater control. We also construct a family of cut-off stochastic pro-cesses adapted to an exhaustion by compact subsets with smooth boundaries, each process is constructed path by path and differentiable in time. Further-more, the differentials have locally uniformly bounded moments with respect to the Brownian motion measures, allowing to bypass the lack of continuity of the exit time of the Brownian motions on its initial position.