Gradient estimates for positive harmonic functions by Stochastic Analysis
by Marc Arnaudon, Bruce K. Driver and Anton Thalmaier

We prove Cheng-Yau type inequalities for positive harmonic functions on Riemannian manifolds by using methods of Stochastic Analysis. Rather than evaluating an exact Bismut formula for the differential of a harmonic function, our method relies on a Bismut type inequality which is derived by an elementary integration by parts argument from an underlying submartingale. It is the monotonicity inherited in this submartingale which allows to establish the pointwise estimates.

Stochastic Processes Appl. 117 (2007) 202-220

The paper is available here:

Marc Arnaudon
Bruce K. Driver
Anton Thalmaier

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