Some remarks on the heat flow for functions and forms
type: Item
instance of: scholarly article
title: Some remarks on the heat flow for functions and forms
zbMATH Open document ID: 0921.58071
author: Anton Thalmaier
published in: Electronic Communications in Probability
publication date: 1998-10-29T00:00:00Z
full work available at URL: https://eudml.org/doc/119935
full work available at URL: paper6.abs.html
review text: This note is concerned with the differentiation of heat semigroups on Riemannian manifolds. In particular, the relation \(dP_tf=P_tdf\) is investigated for the semigroup generated by the Laplacian with Dirichlet boundary conditions. By means of elementary martingale arguments it is shown that well-known properties which hold on complete Riemannian manifolds fail if the manifold is only BM-complete. In general, even if \(M\) is flat and \(f\) smooth of compact support, \(\| dP_tf\|_\infty\) cannot be estimated on compact time intervals in terms of \(f\) or \(df\).
reviewed by: Mihai Gradinaru
Mathematics Subject Classification ID: 58J65
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60H30
zbMATH DE Number: 1215976
zbMATH Keywords: heat semigroup
zbMATH Keywords: heat equation
zbMATH Keywords: Brownian motion
zbMATH Keywords: BM-complete Riemannian manifold
zbMATH Keywords: damped parallel translation
zbMATH Keywords: Ricci curvature
MaRDI profile type: Publication
DOI: 10.1214/ECP.V3-992
description: scientific article; zbMATH DE number 1215976