#### Some remarks on the heat flow for functions
and forms

by Anton Thalmaier

This note is concerned with the differentiation of heat semigroups
on Riemannian manifolds.
In particular, the relation dP_{t}f = P_{t}df
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 dP_{t}f can in general not be estimated
on compact time intervals in terms of f or df.
*Electron. Comm. Probab.*
**3** (1998) 43-49

The paper is also available here:

Anton Thalmaier <anton.thalmaier@uni.lu>
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