Electronic Journal of Probability

Intrinsic Coupling on Riemannian Manifolds and Polyhedra

Max-K. von Renesse

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Starting from a central limit theorem for geometric random walks we give an elementary construction of couplings between Brownian motions on Riemannian manifolds. This approach shows that cut locus phenomena are indeed inessential for Kendall's and Cranston's stochastic proof of gradient estimates for harmonic functions on Riemannian manifolds with lower curvature bounds. Moreover, since the method is based on an asymptotic quadruple inequality and a central limit theorem only it may be extended to certain non smooth spaces which we illustrate by the example of Riemannian polyhedra. Here we also recover the classical heat kernel gradient estimate which is well known from the smooth setting.

Article information

Electron. J. Probab., Volume 9 (2004), paper no. 14, 411-435.

Accepted: 19 April 2004
First available in Project Euclid: 6 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65]
Secondary: 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 58J50: Spectral problems; spectral geometry; scattering theory [See also 35Pxx]

Coupling Gradient Estimates Central Limit Theorem

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von Renesse, Max-K. Intrinsic Coupling on Riemannian Manifolds and Polyhedra. Electron. J. Probab. 9 (2004), paper no. 14, 411--435. doi:10.1214/EJP.v9-205. https://projecteuclid.org/euclid.ejp/1465229700

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