Estimates for transition probabilities on a compact manifold
type: Item
instance of: scholarly article
title: Estimates for transition probabilities on a compact manifold
zbMATH Open document ID: 1110.58029
published in: Journal of Functional Analysis
publication date: 2007-01-16T00:00:00Z
review text: The purpose of this note is to describe a procedure for transferring familiar estimates for transition probabilities on \(\mathbb{R}^N\) to transition probabilities on compact manifolds, maybe filling a gap in the literature. Thus, the author considers, on a compact Riemannian manifold \((M^N, d)\), a uniformly elliptic order two operator \(L\), having Hölder continuous coefficients, and deduces from the well known Euclidean case, under some mild technical conditions, that \(L\) gives rise to a heat equation kernel \(p_r(m,m')\) satisfying Gaussian estimates: \(\exists c>0\), \(\forall t>0\), \(\forall m,m'\in M\), \[ \frac{1}{c\min \{t^{N/2}, 1\}}\exp\left[-\frac ct d(m,m')^2\right]\leq p_t(m,m')\leq\frac{c}{\min \{t^{N/2},1\}}\exp\left[-\frac{d(m,m')^2}{ct}\right]. \]
reviewed by: Jacques Franchi
Mathematics Subject Classification ID: 58J65
zbMATH DE Number: 5116364
zbMATH Keywords: heat kernel
zbMATH Keywords: elliptic operator
zbMATH Keywords: Gaussian estimates
author: Daniel W. Stroock
MaRDI profile type: Publication
full work available at URL: j.jfa.2006.05.008
OpenAlex ID: W2046170774
cites work: A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash
cites work: Q5520962
cites work: Q3862204
DOI: 10.1016/J.JFA.2006.05.008
description: scientific article; zbMATH DE number 5116364