#### Derivative and divergence formulae for diffusion semigroups by Anton Thalmaier and James Thompson

Abstract
For a semigroup $P$tf generated by an elliptic operator on a smooth manifold $M$, we use straightforward martingale arguments to derive probabilistic formulae for $P$t(Vf), not involving derivatives of $f$, where $V$ is a vector field on $M$. For non-symmetric generators, such formulae correspond to the derivative of the heat kernel in the forward variable. As an application, these formulae can be used to derive various shift-Harnack inequalities.

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