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.
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