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Gradient estimates for positive harmonic functions by stochastic analysis |
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| Authors: | Marc Arnaudon Bruce K Driver Anton Thalmaier |
| Institution: | 1. Département de mathématiques, Université de Poitiers, Téléport 2–BP 30179, 86962 Futuroscope Chasseneuil Cedex, France;2. Department of Mathematics-0112, University of California at San Diego, La Jolla, CA 92093-0112, USA;3. Mathematics Laboratory, University of Luxembourg, 162A, Avenue de la Faïencerie, 1511 Luxembourg, Luxembourg |
| 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. |
| Keywords: | 58J65 60H30 |
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