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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

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