Advanced Studies in Pure Mathematics

Li–Yau type gradient estimates and Harnack inequalities by stochastic analysis

Marc Arnaudon and Anton Thalmaier

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Abstract

In this paper we use methods from Stochastic Analysis to establish Li–Yau type estimates for positive solutions of the heat equation. In particular, we want to emphasize that Stochastic Analysis provides natural tools to derive local estimates in the sense that the gradient bound at given point depends only on universal constants and the geometry of the Riemannian manifold locally about this point.

Article information

Source
Probabilistic Approach to Geometry, M. Kotani, M. Hino and T. Kumagai, eds. (Tokyo: Mathematical Society of Japan, 2010), 29-48

Dates
Received: 19 January 2009
Revised: 15 March 2009
First available in Project Euclid: 24 November 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1543086310

Digital Object Identifier
doi:10.2969/aspm/05710029

Mathematical Reviews number (MathSciNet)
MR2605409

Zentralblatt MATH identifier
1201.58023

Subjects
Primary: 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60] 60H30: Applications of stochastic analysis (to PDE, etc.)

Keywords
Heat equation Ricci curvature Li–Yau inequality Harnack inequality gradient bound Brownian motion

Citation

Arnaudon, Marc; Thalmaier, Anton. Li–Yau type gradient estimates and Harnack inequalities by stochastic analysis. Probabilistic Approach to Geometry, 29--48, Mathematical Society of Japan, Tokyo, Japan, 2010. doi:10.2969/aspm/05710029. https://projecteuclid.org/euclid.aspm/1543086310


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