亚椭圆扩散半群的微分
The differentiation of hypoelliptic diffusion semigroups
摘要：
给出了亚椭圆热半群和调和函数的基本导数公式，推广了椭圆情形下的早期工作。重点是在局部鞅的水平上发展分部积分法公式。结合可选的采样定理，这原来是一个有效的方式来处理边界条件，以及有限的生命周期的基础扩散。我们的公式需要hypoellipticity的扩散的意义上的Malliavin演算（可积性的逆Malliavin协方差），并制定在导数流，Malliavin协方差及其逆。最后讨论了调和映射的非线性设定的一些推广。
英文摘要：
Basic derivative formulas are presented for hypoelliptic heat semigroups and harmonic functions extending earlier work in the elliptic case. 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.
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