Basic derivative formulas are presented for hypoelliptic heat semigroups andharmonic functions extending earlier work in the elliptic case. Emphasis isplaced on developing integration by parts formulas at the level of localmartingales. Combined with the optional sampling theorem, this turns out to bean efficient way of dealing with boundary conditions, as well as with finitelifetime of the underlying diffusion. Our formulas require hypoellipticity ofthe diffusion in the sense of Malliavin calculus (integrability of the inverseMalliavin covariance) and are formulated in terms of the derivative flow, theMalliavin covariance and its inverse. Finally some extensions to the nonlinearsetting of harmonic mappings are discussed.
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