On the Differentiation of Heat Semigroups
and Poisson Integrals
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 Hörmander type.
Such formulae provide a direct approach to gradient estimates
for harmonic functions on Riemannian manifolds.
by Anton Thalmaier
Stochastics and Stochastics Reports 61 (1997) 297-321
The paper is available here:
Anton Thalmaier <email@example.com>
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