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

For a semigroup Ptf generated by an elliptic operator on a smooth manifold M, we use straightforward martingale arguments to derive probabilistic formulae for Pt(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.

The paper is available here:

Anton Thalmaier
James Thompson

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