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Gradient estimates for positive harmonic functions by stochastic analysis

Author

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  • Arnaudon, Marc
  • Driver, Bruce K.
  • Thalmaier, Anton

Abstract

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 us to establish the pointwise estimates.

Suggested Citation

  • Arnaudon, Marc & Driver, Bruce K. & Thalmaier, Anton, 2007. "Gradient estimates for positive harmonic functions by stochastic analysis," Stochastic Processes and their Applications, Elsevier, vol. 117(2), pages 202-220, February.
    • Handle: RePEc:eee:spapps:v:117:y:2007:i:2:p:202-220
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    Cited by:

    1. Chen, Xin & Cheng, Li-Juan & Mao, Jing, 2015. "A probabilistic method for gradient estimates of some geometric flows," Stochastic Processes and their Applications, Elsevier, vol. 125(6), pages 2295-2315.
    2. Cheng, Li-Juan, 2014. "A probabilistic approach for gradient estimates on time-inhomogeneous manifolds," Statistics & Probability Letters, Elsevier, vol. 88(C), pages 174-183.

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