Some remarks on the heat flow for functions
This note is concerned with the differentiation of heat semigroups
on Riemannian manifolds.
In particular, the relation dPtf = Ptdf
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.
Even if M is flat and f smooth of compact support,
the infinity norm of dPtf can in general not be estimated
on compact time intervals in terms of f or df.
by Anton Thalmaier
Electron. Comm. Probab.
3 (1998) 43-49
The paper is also available here:
Anton Thalmaier <email@example.com>
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