The Differentiation of Hypoelliptic Diffusion Semigroups
نویسنده
چکیده
Basic derivative formulas are presented for hypoelliptic heat semi-groups and harmonic functions extending earlier work in the elliptic case. Following the approach of 15], 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 eecient way of dealing with boundary conditions, as well as with nite lifetime of the underlying diiusion. Our formulas require hypoellipticity of the diiusion in the sense of Malliavin calculus (integrability of the inverse Malliavin covariance) and are formulated in terms of the derivative ow, the Malliavin covariance and its inverse. See 7] for related results on integration by parts for degenerate diiusion measures. Finally some extensions to the nonlinear setting of harmonic mappings are discussed.
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