DERIVATIVE AND DIVERGENCE FORMULAE FOR DIFFUSION SEMIGROUPS
ANTON THALMAIER AND JAMES THOMPSON
Mathematics Research Unit, FSTC, University of Luxembourg 6, rue Richard Coudenhove-Kalergi, 1359 Luxembourg, Grand Duchy of Luxembourg
Abstract. 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 non-symmetric genera- tors, 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.
Introduction
For a Banach space E, e ∈ E and a Markov operator P on Bb(E), it is known that certain estimates on P(∇e f ) are equivalent to corresponding shift-Harnack inequalities. This was proved by F.-Y. Wang in [18]. For example, for δe ∈ (0,1) and βe ∈ C((δe,∞) × E;[0,∞)), he proved that the derivative-entropy estimate