Derivative and divergence formulae for diffusion semigroups.

For a semigroup Pt generated by an elliptic operator on a smooth manifold M, we use straightforward martingale arguments to derive probabilistic formulae for Pt(V(f)), not involving derivatives of f, where V is a vector field on M. For nonsymmetric 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.