Abstract
For a semigroup $P_{t}$ generated by an elliptic operator on a smooth manifold $M$, we use straightforward martingale arguments to derive probabilistic formulae for $P_{t}(V(f))$, 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.
Citation
Anton Thalmaier. James Thompson. "Derivative and divergence formulae for diffusion semigroups." Ann. Probab. 47 (2) 743 - 773, March 2019. https://doi.org/10.1214/18-AOP1269
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