On the differentiation of heat semigroups and poisson integrals

Abstract

We give a version of integration by parts on the level of local martingales; combined with the optional sampling theorem, this method allows us to obtain differentiation formulae for Poisson integrals in the same way as for heat semigroups involving boundary conditions. In particular, our results yield Bismut type representations for the logarithmic derivative of the Poisson kernel on regular domains in Riemannian manifolds corresponding to elliptic PDOs of Hormander type. Such formulae provide a direct approach to gradient estimates for harmonic functions on Riemannian manifolds

Keywords:

• Diffusion
• heat semigroup
• integration by parts
• heat kernel; Poisson kernel
• 1991 Mathematics Subject Classification: 58G32, 60H10, 60H30