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1998 Some Remarks on the Heat Flow for Functions and Forms
Anton Thalmaier
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Electron. Commun. Probab. 3: 43-49 (1998). DOI: 10.1214/ECP.v3-992

Abstract

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, $\Vert dP_tf\Vert_\infty$ cannot be estimated on compact time intervals in terms of $f$ or $df$.

Citation

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Anton Thalmaier. "Some Remarks on the Heat Flow for Functions and Forms." Electron. Commun. Probab. 3 43 - 49, 1998. https://doi.org/10.1214/ECP.v3-992

Information

Accepted: 18 July 1998; Published: 1998
First available in Project Euclid: 2 March 2016

zbMATH: 0921.58071
MathSciNet: MR1637977
Digital Object Identifier: 10.1214/ECP.v3-992

Subjects:
Primary: 58G32
Secondary: 60H10, 60H30

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