The Annals of Probability

Derivative and divergence formulae for diffusion semigroups

Anton Thalmaier and James Thompson

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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.

Article information

Ann. Probab., Volume 47, Number 2 (2019), 743-773.

Received: January 2017
Revised: February 2018
First available in Project Euclid: 26 February 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60] 60J60: Diffusion processes [See also 58J65]
Secondary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]

Diffusion semigroup heat kernel gradient estimate Harnack inequality Ricci curvature


Thalmaier, Anton; Thompson, James. Derivative and divergence formulae for diffusion semigroups. Ann. Probab. 47 (2019), no. 2, 743--773. doi:10.1214/18-AOP1269.

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