Derivative and divergence formulae
for diffusion semigroups
by Anton Thalmaier and James Thompson
Abstract
For a semigroup generated by an elliptic operator on a smooth
manifold , we use straightforward martingale arguments to derive
probabilistic formulae for , not involving derivatives of
, where is a vector field on . 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.
Ann. Probab. 47 (2019), no. 2, 743-773
http://dx.doi.org/10.1214/18-AOP1269
The paper is available here:
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