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 Electronic Communications in Probability > Vol. 3 (1998) > Paper 6 open journal systems 


Some Remarks on the Heat Flow for Functions and Forms

Anton Thalmaier, University of Bonn


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, $Vert dP_tfVert_infty$ cannot be estimated on compact time intervals in terms of $f$ or $df$.


Full text: PDF

Pages: 43-49

Published on: July 18, 1998
Bibliography
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Electronic Communications in Probability. ISSN: 1083-589X