#### The Differentiation of
Hypoelliptic Diffusion Semigroups

by Marc Arnaudon and Anton Thalmaier

Basic derivative formulas are presented for hypoelliptic heat semigroups and
harmonic functions extending earlier work in the elliptic case. Following
the approach of Thalmaier (1997), emphasis is placed on developing integration
by parts formulas at the level of local martingales. Combined with the
optional sampling theorem, this turns out to be an efficient way of dealing
with boundary conditions, as well as with finite lifetime of the underlying
diffusion. Our formulas require hypoellipticity of the diffusion in the
sense of Malliavin calculus (integrability of the inverse Malliavin
covariance) and are formulated in terms of the derivative flow, the
Malliavin covariance and its inverse.
Finally some extensions to the nonlinear setting of harmonic mappings are
discussed.

*llinois J. Math. ***54** (2010) 1285-1311

The paper is available here:

Marc Arnaudon marc.arnaudon@math.univ-poitiers.fr
Anton Thalmaier
anton.thalmaier@uni.lu
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