 
Summary: THE DIFFERENTIATION OF HYPOELLIPTIC DIFFUSION
SEMIGROUPS
MARC ARNAUDON AND ANTON THALMAIER
Dedicated to the memory of Donald Burkholder
Abstract. Basic derivative formulas are presented for hypoelliptic heat semi
groups and harmonic functions extending earlier work in the elliptic case. Fol
lowing the approach of [22], 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 bound
ary conditions, as well as with finite lifetime of the underlying diffusion. Our
formulas require hypoellipticity of the diffusion in the sense of Malliavin cal
culus (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.
Contents
1. Introduction 1
2. Hypoellipticity and the Malliavin Covariance 3
3. A Basic Integration by Parts Argument 5
4. Integration by Parts at the Level of Local Martingales 7
5. Hypoelliptic Diffusions and Control Theory 8
